Academic Publications
Publications
My publications are listed at my ORCID page.
Many of my later papers can be found on my arXiv page.
See also my Google Scholar page
Preprints
Chapters in Books
Aleksander L. Owczarek and Stuart G. Whittington
Interacting Lattice Polygons
Polygons, Polyominoes and Polycubes (A. J. Guttmann (Ed.)) Lecture Notes in Physics vol 775 (Springer/Canopus (Bristol)) pp 301–16 (2009)
Link | PDF file
Conference Proceedings (refereed)
Nicholas R. Beaton, Aleksander L. Owczarek, and Ruijie Xu
Quarter-plane lattice paths with interacting boundaries: the Kreweras and reverse Kreweras models
Transcendence in Algebra, Combinatorics, Geometry and Number Theory (Proceedings of Trans'19)
Springer Proceedings in Mathematics & Statistics, Volume 373 (2021), 163-192.
Link | arXiv |
Show/Hide Abstract | Mathematica files
Lattice paths in the quarter plane have led to a large and varied set of results in recent years.
One major project has been the classification of step sets according to the properties of the corresponding generating functions, and this has involved a variety of techniques,
some highly intricate and specialised. The famous Kreweras and reverse Kreweras walk models are two particularly interesting models,
as they are among the only four cases which have algebraic generating functions.
Here we investigate how the properties of the Kreweras and reverse Kreweras models change when boundary interactions are introduced.
That is, we associate three real-valued weights a,b,c with visits by the walks to the x-axis, the y-axis and the origin (0,0) respectively.
These models were partially solved in a recent paper by Beaton, Owczarek and Rechnitzer (2019). We apply the algebraic kernel method to completely solve these two models.
We find that reverse Kreweras walks have an algebraic generating function for all a,b,c,
regardless of whether the walks are restricted to end at the origin or on one of the axes, or may end anywhere at all. For Kreweras walks,
the generating function for walks returning to the origin is algebraic, but the other cases are only D-finite.
We also compare these two models with the other solvable cases, and observe that they have some unique properties.
Nicholas R. Beaton, Aleksander L. Owczarek, and Ruijie Xu
Quarter-plane lattice paths with interacting boundaries: Kreweras and friends
Proceedings of the 31st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2019, Ljubljana, Slovenia)
Link | PDF file |
Show/Hide Abstract | Mathematica file
We study lattice paths in the quarter-plane which accrue weights with each visit to the $x$-axis, the $y$-axis and the origin.
In particular, we address two cases which were only partially solved in a recent work by Beaton, Owczarek and Rechnitzer (2018):
Kreweras and reverse Kreweras paths. Without weights these are two of the famous algebraic quarter-plane models.
We show that the reverse Kreweras model remains algebraic for all possible weights,
while the nature of the Kreweras model appears to depend on the value of the weights and the endpoint of the paths.
Arturo Narros, Aleksander L. Owczarek and Thomas Prellberg
Winding angle distributions for two-dimensional collapsing polymers
Journal of Physics: Conference Series 686 012007 (9 pages) (2016)
Link | PDF file |
arXiv |
Show/Hide Abstract
We provide numerical support for a long-standing
prediction of universal scaling of winding angle distributions.
Simulations of interacting self-avoiding walks show that the winding angle distribution for N-step walks
is compatible with the theoretical prediction of a Gaussian with a variance growing asymptotically as C log N,
with C = 2 in the swollen phase (previously verified), and C = 24/7 at the θ-point. At low temperatures weaker evidence
demonstrates compatibility with the same scaling and a value of C = 4 in the collapsed phase, also as theoretically predicted.
Aleksander L. Owczarek and Andrew Rechnitzer
On the number of anisospiral walks: a challenge in numerical analysis
Counting Complexity: An International Workshop On Statistical Mechanics And Combinatorics
Journal of Physics: Conference Series vol 42 pp 225–30 (2006)
Link | PDF file |
Show/Hide Abstract
The numerical analysis of combinatorial problems with non-standard scaling is an important testing ground for the limits of
current techniques. One problem that has proven especially difficult to analyse with all available numerical techniques,
including various Monte Carlo simulation methods and careful series analysis, is anisotropic spiral walks in two dimensions.
Here we revisit this problem discussing various non-standard scaling hypotheses and showing how these best fit the available data.
This highlights the difficulties with the analysis of data when the standard scaling forms may not hold true and also provides a
testing ground for improved techniques.
Richard Brak, John Essam, Judy-Anne Osborn, Aleksander L. Owczarek and Andrew Rechnitzer
Lattice Paths and the Constant Term
Counting Complexity: An International Workshop On Statistical Mechanics And Combinatorics
Journal of Physics: Conference Series vol 42 pp 47–58 (2006)
Link | PDF file |
Show/Hide Abstract
We firstly review the constant term method (CTM),
illustrating its combinatorial connections and show how it can be used to solve a certain class of lattice path problems.
We show the connection between the CTM, the transfer matrix method (eigenvectors and eigenvalues), partial difference equations,
the Bethe Ansatz and orthogonal polynomials. Secondly, we solve a lattice path problem first posed in 1971.
The model stated in 1971 was only solved for a special case - we solve the full model.
Aleksander L. Owczarek
A note on the corrections-to-scaling for the number of nearest neighbour contacts in self-avoiding walks
Counting Complexity: An International Workshop On Statistical Mechanics And Combinatorics
Journal of Physics: Conference Series vol 42 pp 221–4 (2006)
Link | PDF file |
Show/Hide Abstract
Recently, an exhaustive study has been made of the corrections-to-scaling for the number of, and various size measures
(eg. radius of gyration) of, self-avoiding walks on the various two-dimensional lattices. This study gave compelling evidence
that the first non-analytic correction-to-scaling has exponent △1 = 3⁄2. However, there also exist predictions in the literature
for the corrections-to-scaling of the number of nearest neighbour contacts of self-avoiding walks. These are partially based on the
analysis of relatively short series. Here we demonstrate that the form for the scaling of the number of self-avoiding walks recently proposed,
and some standard scaling assumptions, implies that this older conjecture on the corrections-to-scaling for the number of
nearest neighbour contacts is unlikely to hold. We consolidate this claim by the analysis of Monte Carlo data for both
two and three dimensional self-avoiding walks.
This work also shows that the often standard assumption that all quantities have the same corrections-to-scaling is misleading.
Thomas Prellberg and Aleksander L. Owczarek
Polymer Collapse in High Dimensions: Monte Carlo Simulation of Lattice Models
Computer Simulation Studies in Condensed Matter Physics XVI (Springer Verlag) Vol 95 pp
147-151 (2004)
Link | PDF file |
Show/Hide Abstract
We present Monte Carlo simulations of the coil-globule transition for interacting
self-avoiding walks (ISAW) and interacting self-avoiding trails (ISAT) on the hyper-cubic lattice in four and five dimensions,
performed with the PERM algorithm. We find that the second-order nature of the coil-globule transition is masked by pseudo-first-order behaviour,
i.e. the build-up of first-order-like singularities due to strong finite-size corrections to scaling.
Richard Brak, John Essam and Aleksander L. Owczarek
Equivalence of the Bethe Ansatz and the Gessel-Viennot Theorem for Non-intersecting Paths
Proceedings of the 11th Formal Power Series and Algebraic Combinatorics Conference pp 108–18 (1999)
PDF file |
Show/Hide Abstract
We show how the problem of non-intersecting lattice paths
on the directed square lattice can be formulated as
difference equations. The difference equations are
encoded by the action of various ``transfer matrices''.
We state several theorems that demonstrate how the
coordinate Bethe Ansatz for the eigenvectors of the
transfer matrices, given certain conditions hold, is
equivalent to the Gessel-Viennot determinant for the
number of configurations of $N$ non-intersecting lattice
paths on the directed square lattice. Another way of
viewing this result is that it is a linear algebra proof
of the Gessel-Viennot theorem for the particular case
considered in this paper. This is significant as the
Bethe Ansatz is potentially capable of solving various
lattice paths problems, such as osculating lattice paths,
which are beyond the scope of the Gessel-Viennot theorem.
Thomas Prellberg and Aleksander L. Owczarek
Partially Convex Lattice Vesicles: Methods and Recent Results
Proceedings of the Conference `Confronting the Infinite’ (World Scientific) pp 204–14 (1995)
Link | PDF file |
Show/Hide Abstract
We review the methods and results achieved recently
for some lattice models of vesicles in two dimensions, which are
defined by the restriction of partial convexity. These models provide
an alternate testing ground for the scaling and universality
hypotheses to the more intricate Ising model. The scaling functions
can be calculated, in addition to the exponents, and we conclude that
all the most complex models fall into one universality class. We also
present, as a pedagogical example of the methods, the scaling
behaviour of a model not previously studied in this fashion.
Journal Articles (refereed)
Chris J. Bradly, Nicholas R. Beaton and Aleksander L. Owczarek
Lattice polymers near a permeable interface
Journal of Physics A: Mathematical and Theoretical 57 445004 (17 pp) (2024)
Link | PDF file
| arXiv |
Show/Hide Abstract
We study the localisation of lattice polymer models near a permeable interface in two dimensions.
Localisation can arise due to an interaction between the polymer and the interface, and can be altered by a preference for the bulk solvent on one side
or by the application of a force to manipulate the polymer. Different combinations of these three effects give slightly different statistical mechanical behaviours.
The canonical lattice model of polymers is the self-avoiding walk which we mainly study with Monte Carlo simulation to calculate the phase diagram and critical phenomena.
For comparison, a solvable directed walk version is also defined and the phase diagrams are compared for each case. We find broad agreement between the two models,
and most minor differences can be understood as due to the different entropic contributions.
In the limit where the bulk solvent on one side is overwhelmingly preferred we see how the localisation transition transforms to the adsorption transition;
the permeable interface becomes effectively an impermeable surface.
Nicholas R. Beaton and Aleksander L. Owczarek
Exact solution of weighted partially directed walks crossing a square
Journal of Physics A: Mathematical and Theoretical 56 155003 (20 pp) (2023)
Link | PDF file
| arXiv |
Show/Hide Abstract
We consider partially directed walks crossing a $L\times L$ square weighted according to their length by a fugacity $t$.
The exact solution of this model is computed in three different ways, depending on whether $t$ is less than, equal to or greater than 1.
In all cases a complete expression for the dominant asymptotic behaviour of the partition function is calculated.
The model admits a dilute to dense phase transition, where for $0 < t < 1$ the partition function
scales exponentially in $L$ whereas for $t>1$ the partition function scales exponentially in $L^2$,
and when $t=1$ there is an intermediate scaling which is exponential in $L \log{L}$.
As such we provide an exact solution of a model of the dilute to dense polymeric phase transition in two dimensions.
Chris Bradly and Aleksander L. Owczarek
Critical behaviour of the extended-ballistic transition for pulled self-avoiding walks
Physica A 624 128978 (8 pp) (2023)
Link | PDF file |
Show/Hide Abstract
In order to study long chain polymers many lattice models
accommodate a pulling force applied to a particular part of the chain, often a free endpoint.
This is in addition to well-studied features such as energetic interaction between the lattice polymer and a surface.
However, the critical behaviour of the pulling force alone is less well studied, such as characterising the nature of the
phase transition and particularly the values of the associated exponents. We investigate a simple model of lattice polymers
subject to forced extension, namely self-avoiding walks (SAWs) on the square and simple cubic lattices with one endpoint attached
to an impermeable surface and a force applied to the other endpoint acting perpendicular to the surface.
In the thermodynamic limit the system undergoes a transition to a ballistic phase as the force is varied and
it is known that this transition occurs whenever the magnitude of the force is positive, i.e.f > f_c=0.
Using well established scaling arguments we show that the crossover exponent phi
for the finite-size model is identical to the well-known exponent nu_d, which controls the scaling of the size of the polymer in
d-dimensions. With extensive Monte Carlo simulations we test this conjecture and show that the value of phi
is indeed consistent with the known values of nu_2=3/4
and nu_3 =0.587597(7). Scaling arguments, in turn, imply the specific heat exponent alpha
is 2/3
in two dimensions and 0.29815(2)
in three dimensions.
Chris Bradly and Aleksander L. Owczarek
Polymer collapse of a self-avoiding trail model on a two-dimensional inhomogeneous lattice
Physica A 604 127688 (11pp) (2022)
Link | PDF file |
Show/Hide Abstract
The study of the effect of random impurities on the collapse of a flexible polymer
in dilute solution has had recent attention with consideration of semi-stiff interacting self-avoiding walks on the square lattice.
In the absence of impurities the model displays two types of collapsed phase, one of which is both anisotropically ordered and maximally dense
(crystal-like). In the presence of impurities the study showed that the crystal type phase disappears. Here we investigate
extended interacting self-avoiding trails on the triangular lattice with random impurities. Without impurities this model
also displays two collapsed phases, one of which is maximally dense but not ordered anisotropically and so may be affected
differently by a disorderd lattice. The trails are simulated using the flatPERM algorithm and the inhomogeneity is
realised by making a random fraction of the lattice unavailable to the trails. We calculate several thermodynamic and
metric quantities to map out the phase diagram and look at how the amount of disorder affects the properties of each phase
but especially the maximally dense phase. Our results indicate that while the maximally dense phase in the trail model is
affected less than in the walk model it is also disrupted and becomes a denser version of the globule phase so that the model
with impurities displays no more than one true thermodynamic collapsed phase.
Chris Bradly and Aleksander L. Owczarek
Critical scaling of lattice polymers confined to a box without endpoint restriction
J. Chem. Phys 60 (18pp) (2022)
Link | PDF file |
Show/Hide Abstract
We study self-avoiding walks on the square lattice restricted to a
square box of side L weighted by a length fugacity without restriction of their end points.
This is a natural model of a confined polymer in dilute solution such as polymers in mesoscopic pores.
The model admits a phase transition between an ‘empty’ phase, where the average length of walks are finite and the
density inside large boxes goes to zero, to a ‘dense’ phase, where there is a finite positive density.
We prove various bounds on the free energy and develop a scaling theory for the phase transition based on the
standard theory for unconstrained polymers. We compare this model to unrestricted walks and walks that whose
endpoints are fixed at the opposite corners of a box, as well as Hamiltonian walks. We use Monte Carlo simulations
to verify predicted values for three key exponents: the density exponent alpha =1/2, the finite size crossover exponent
1/nu=4/3 and the critical partition function exponent 2-eta = 43/32.
This implies that the theoretical framework relating them to the unconstrained SAW problem is valid.
Anthony J. Guttmann, Iwan Jensen and Aleksander L. Owczarek
Self-avoiding walks contained within a square
J. Phys. A: Math. Theor. 55 415201(16pp) (2022)
Link | PDF file |
Show/Hide Abstract
We have studied self-avoiding walks contained within an L × L square
whose end-points can lie anywhere within, or on, the boundaries of the square. We prove that such walks behave,
asymptotically, as walks crossing a square (WCAS), being those walks whose end-points lie at the south-east and north-west
corners of the square. We provide numerical data, enumerating all such walks, and analyse the sequence of coefficients
in order to estimate the asymptotic behaviour. We also studied a subset of these walks, those that must contain at least
one edge on all four boundaries of the square. We provide compelling evidence that these two classes of walks grow identically.
From our analysis we conjecture that the number of such walks CL, for both problems, behaves as
C_L \sim lambda^{L^2+bL+c} L^g,
where (Guttmann and Jensen 2022 J. Phys. A: Math. Theor.) λ = 1.744 5498 ± 0.000 0012,
b = −0.043 54 ± 0.0005, c = −1.35 ± 0.45, and g = 3.9 ± 0.1. Finally, we also studied
the equivalent problem for self-avoiding polygons, also known as cycles in a square grid.
The asymptotic behaviour of cycles has the same form as walks, but with different values of the parameters c, and g.
Our numerical analysis shows that λ and b have the same values as for WCAS and that c = 1.776 ± 0.002 while g = −0.500 ± 0.005
and hence probably equals -1/2.
Nicholas R. Beaton and Aleksander L. Owczarek
Exact solutions of directed walk models of polymeric zipping with pulling in two and three dimensions
Physica A: Statistical Mechanics and its Applications 566 (2021) 125635.
Link | PDF file
| arXiv |
Show/Hide Abstract
We provide the exact solution of several variants of simple models of the zipping transition of two bound polymers, such as occurs in DNA/RNA,
in two and three dimensions using pairs of directed lattice paths. In three dimensions the solutions are written in terms of complete elliptic integrals.
We analyse the phase transition associated with each model giving the scaling of the partition function.
We also extend the models to include a pulling force between one end of the pair of paths,
which competes with the attractive monomer-monomer interactions between the polymers.
Chris Bradly and Aleksander L. Owczarek
Effect of lattice inhomogeneity on collapsed phases of semi-stiff ISAW polymers
Journal of Statistical Physics 182 27 (12 pages) (2021)
Link | PDF file
| arXiv |
Show/Hide Abstract
We investigate semi-stiff interacting self-avoiding walks on the square lattice with random impurities.
The walks are simulated using the flatPERM algorithm and the inhomogeneity is realised as a random fraction of the lattice that is unavailable to the walks.
We calculate several thermodynamic and metric quantities to map out the phase diagram and look at how the amount of disorder affects the properties of each phase.
On a homogeneous lattice this model has an extended phase and two distinct collapsed phases, globular and crystalline, which differ in the anisotropy of the walks.
By adding impurities to the lattice we notice a degree of swelling of the walks for all phases that is commensurate with the fraction of the lattice that is removed.
Importantly, the crystal phase disappears with the addition of impurities for sufficiently long walks. For finite length walks we demonstrate that competition between
the size of the average spaces free of impurities and the size of the collapsed polymer describes the crossover between the homogeneous lattice
and the impurity dominated situation.
Nicholas R. Beaton, Aleksander L. Owczarek, and Andrew Rechnitzer
Exact solution of some quarter plane walks with interacting boundaries
Electronic Journal of Combinatorics 26(3) (38 pp)(2019)
Link | PDF file |
arXiv |
Show/Hide Abstract
The set of random walks with different step sets (of short steps) in the quarter plane has
provided a rich set of models that have profoundly different integrability properties. In particular, 23 of the 79 effectively different models
can be shown to have generating functions that are algebraic or differentiably finite. Here we investigate how this integrability may change in those 23
models where in addition to length one also counts the number of sites of the walk touching either the horizontal and/or vertical boundaries of the quarter plane.
This is equivalent to introducing interactions with those boundaries in a statistical mechanical context.
We are able to solve for the generating function in a number of cases.
For example, when counting the total number of boundary sites without differentiating whether they are horizontal or vertical,
we can solve the generating function of a generalised Kreweras model.
However, in many instances we are not able to solve as the kernel methodology seems to break down when including counts with the boundaries.
Aleksander L. Owczarek and Thomas Prellberg
Exact solution of pulled, directed vesicles with sticky walls in two dimensions
J. Math. Phys 60 033301 (8pp)(2019)
Link | PDF file
Show/Hide Abstract
We analyse a directed lattice vesicle model incorporating
both the binding-unbinding transition and the vesicle inflation-deflation transition. From the exact solution,
we derive the phase diagram for this model and elucidate scaling properties around the binding-unbinding critical
point in this larger parameter space. We also consider how the
phase diagram changes when a perpendicular force is applied to the end of a directed vesicle.
Chris Bradly and Aleksander L. Owczarek
Monte Carlo simulation of 3-star lattice polymers pulled from an adsorbing surface
J. Phys. A: Math. Theor. 52 275001 (15pp)(2019)
Link | PDF file |
arXiv |
Show/Hide Abstract
We study uniform 3-star polymers with one branch tethered
to an attractive surface and another branch pulled by a force away from the surface.
Each branch of the 3-star lattice is modelled as a self-avoiding walk (SAW) on the simple cubic lattice
with one endpoint of each branch joined at a common node. Recent theoretical work van Rensburg and Whittington
(2018 J. Phys. A: Math. Theor. 51 204001) found four phases for this system: free, fully adsorbed, ballistic and mixed.
The mixed phase occurs between the ballistic and fully adsorbed phase.
We investigate this system by using the flatPERM Monte Carlo algorithm with special restrictions
on the endpoint moves to simulate 3-stars up to branch length 128. We provide numerical evidence of the
four phases and in particular that the ballistic-mixed and adsorbed-mixed phase boundaries are first-order transitions.
The position of the ballistic-mixed and adsorbed-mixed boundaries are found at the expected location in the asymptotic
regime of large force and large surface-monomer interaction energy.
These results indicate that the flatPERM algorithm is suitable for simulating star lattice polymers
and opens up new avenues for numerical study of non-linear lattice polymers.
Chris Bradly, Aleksander L. Owczarek and Thomas Prellberg
Phase transitions in solvent dependent polymer adsorption in three dimensions
Phys. Rev. E 99 062113 (5pp) (2019)
Link | PDF file |
arXiv |
Show/Hide Abstract
We consider the phase diagram of self-avoiding walks (SAWs) on the
simple cubic lattice subject to surface and bulk interactions, modeling an adsorbing surface and variable
solvent quality for a polymer in dilute solution, respectively. We simulate SAWs at specific interaction
strengths to focus on locating certain transitions and their critical behavior. By collating these new
results with previous results we sketch the complete phase diagram and show how the adsorption transition
is affected by changing the bulk interaction strength.
This expands on recent work considering how adsorption is affected by solvent quality. We demonstrate that
changes in the adsorption crossover exponent coincide with phase boundaries.
Chris Bradly, Buks van Rensburg, Aleksander L. Owczarek and Stu G. Whittington
Force-induced desorption of 3-star polymers in two dimensions
J. Phys. A: Math. Theor. 52 315002 (13pp) (2019)
Link | PDF file |
arXiv |
Show/Hide Abstract
We investigate the phase diagram of a self-avoiding walk model of a
3-star polymer in two dimensions, adsorbing at a surface and being desorbed by the action of a force.
We show rigorously that there are four phases: a free phase, a ballistic phase,
an adsorbed phase and a mixed phase where part of the 3-star is adsorbed and part is ballistic.
We use both rigorous arguments and Monte Carlo methods to map out the phase diagram, and
investigate the location and nature of the phase transition boundaries.
In two dimensions, only two of the arms can be fully adsorbed in the surface
and this alters the phase diagram when compared to 3-stars in three dimensions.
Nathann T. Rodrigues, Thomas Prellberg and Aleksander L. Owczarek
Adsorption of interacting self-avoiding trails in two dimensions
Phys. Rev. E 100 022121 (10pp) (2019)
Link | PDF file |
arXiv |
Show/Hide Abstract
We investigate the surface adsorption transition of interacting
self-avoiding square lattice trails onto a straight boundary line. The character of this adsorption transition
depends on the strength of the bulk interaction, which induces a collapse transition of the trails from a
swollen to a collapsed phase, separated by a critical state. If the trail is in the critical state,
the universality class of the adsorption transition changes; this is known as the special adsorption point.
Using flatPERM, a stochastic growth Monte Carlo algorithm, we simulate the adsorption of self-avoiding
interacting trails on the square lattice using three different boundary scenarios which differ with
respect to the orientation of the boundary and the type of surface interaction. We confirm the expected phase diagram,
showing swollen, collapsed, and adsorbed phases in all three scenarios, and confirm universality
of the normal adsorption transition at low values of the bulk interaction strength.
Intriguingly, we cannot confirm universality of the special adsorption transition.
We find different values for the exponents; the most likely explanation is that this
is due to the presence of strong corrections to scaling at this point.
Chris Bradly, Buks van Rensburg, Aleksander L. Owczarek and Stu G. Whittington
Adsorbed self-avoiding walk pulled at an interior vertex
J. Phys. A: Math. Theor. 52 405001(14pp) (2019)
Link | PDF file |
arXiv |
Show/Hide Abstract
We consider self-avoiding walks terminally attached to a surface
at which they can adsorb. A force is applied, normal to the surface, to desorb the walk and we investigate how
the behaviour depends on the vertex of the walk at which the force is applied. We use rigorous arguments to map
out some features of the phase diagram, including bounds on the locations of some phase boundaries,
and we use Monte Carlo
methods to make quantitative predictions about the locations of these boundaries
and the nature of the various phase transitions.
Nathann T. Rodrigues, Tiago J. Oliveira, Thomas Prellberg and Aleksander L. Owczarek
Adsorption of two-dimensional polymers with two- and three-body self-interaction
Phys. Rev. E 100 062504 (11pp) (2019)
Link | PDF file |
arXiv |
Show/Hide Abstract
Using extensive Monte Carlo simulations,
we investigate the surface adsorption of self-avoiding trails on the triangular lattice with
two- and three-body on-site monomer-monomer interactions. In the parameter space of two-body, three-body,
and surface interaction strengths, the phase diagram displays four phases: swollen (coil), globule, crystal,
and adsorbed. For small values of the surface interaction, we confirm the presence of swollen, globule,
and crystal bulk phases. For sufficiently large values of the surface interaction, the system is in an adsorbed state,
and the adsorption transition can be continuous or discontinuous, depending on the bulk phase.
As such, the phase diagram contains a rich phase structure with transition surfaces that meet in
multicritical lines joining in a single special multicritical point.
The adsorbed phase displays two distinct regions with different characteristics,
dominated by either single- or double-layer adsorbed ground states.
Interestingly, we find that there is no finite-temperature phase transition
between these two regions though rather a smooth crossover.
Arturo Narros, Aleksander L. Owczarek and Thomas Prellberg
Anomalous polymer collapse winding angle distributions
J. Phys. A: Math. Theor. 51 114001 (9pp) (2018)
Link | PDF file |
arXiv |
Show/Hide Abstract
In two dimensions polymer collapse has been shown to be complex with multiple low temperature states and multi-critical points.
Recently, strong numerical evidence has been provided for a long-standing prediction of universal scaling of winding angle distributions,
where simulations of interacting self-avoiding walks show that the winding angle distribution for N-step walks is compatible with the
theoretical prediction of a Gaussian with a variance growing asymptotically as C log N. Here we extend this work by considering interacting
self-avoiding trails which are believed to be a model representative of some of the more complex behaviour. We provide robust evidence that, while the high temperature swollen state of this model has a winding angle distribution that is also Gaussian, this breaks down at the polymer collapse point and at low temperatures. Moreover, we provide some evidence that the distributions are well modelled by stretched/compressed exponentials, in contradistinction to the behaviour found in interacting self-avoiding walks.
Chris Bradly, Aleksander L. Owczarek and Thomas Prellberg
Universality of crossover scaling for the adsorption transition of lattice polymers
Phys. Rev. E 97 022503 (10pp) (2018)
Link | PDF file |
arXiv |
Show/Hide Abstract
Recently, it has been proposed that the adsorption transition for a
single polymer in dilute solution, modeled by lattice walks in three dimensions, is not universal with respect
to intermonomer interactions. Moreover, it has been conjectured that key critical exponents 𝜙, measuring the
growth of the contacts with the surface at the adsorption point, and 1/𝛿, which measures the finite-size shift
of the critical temperature, are not the same. However, applying standard scaling arguments the two key critical
exponents should rather be identical, hence pointing to a potential breakdown of these standard scaling arguments.
Both of these conjectures are in contrast to the well-studied situation in two dimensions, where there are exact
results from conformal field theory: these exponents are both accepted to be 1/2 and universal. We use the flatPERM
algorithm to simulate self-avoiding walks and trails on the hexagonal, square, and simple cubic lattices up to length
1024 to investigate these claims. Walks can be seen as a repulsive limit of intermonomer interaction for trails,
allowing us to probe the universality of adsorption. For each lattice model we analyze several thermodynamic properties
to produce different methods of estimating the critical temperature and the key exponents. We test our methodology
on the two-dimensional cases, and the resulting spread in values for 𝜙 and 1/𝛿 indicates that there is a systematic
error which can far exceed the statistical error usually reported. We further suggest a methodology for consistent
estimation of the key adsorption exponents which gives 𝜙=1/𝛿=0.484(4) in three dimensions. Hence, we conclude that
in three dimensions these critical exponents indeed differ from the mean-field value of 1/2, as had previously been
calculated,
but cannot find evidence that they differ from each other. Importantly, we also find no substantive evidence of any
nonuniversality in the polymer adsorption transition.
Chris Bradly, Aleksander L. Owczarek and Thomas Prellberg
Adsorption of neighbor-avoiding walks on the simple cubic lattice
Phys. Rev. E 98 062141 (6pp) (2018)
Link | PDF file |
arXiv |
Show/Hide Abstract
We investigate neighbor-avoiding walks on the simple cubic lattice
in the presence of an adsorbing surface. This class of lattice paths has been less studied using Monte Carlo
simulations. Our investigation follows on from our previous results using self-avoiding walks and self-avoiding trails.
The connection is that neighbor-avoiding walks are equivalent to the infinitely repulsive limit of self-avoiding walks
with monomer-monomer interactions. Such repulsive interactions can be seen to enhance the excluded volume effect.
We calculate the critical behavior of the adsorption transition for neighbor-avoiding walks,
finding a critical temperature of 𝑇a=3.274(9) and a crossover exponent of 𝜙=0.482(13), w
hich is consistent with the exponent for self-avoiding walks and trails,
leading to an overall combined estimate for three dimensions of 𝜙3D=0.484(7).
While questions of universality have previously been raised regarding the value of
adsorption exponents in three dimensions, our results indicate that the
value of 𝜙 in the strongly repulsive regime does not differ from its noninteracting value.
However, it is clearly different from the mean-field value of 1/2 and therefore not superuniversal.
Andrea Bedini, Aleksander L. Owczarek and Thomas Prellberg
Self-attracting polymers in two dimensions with three low-temperature phases
J. Phys. A: Math. Theor. 50 095003 (28pp) (2017)
Link | PDF file |
arXiv |
Show/Hide Abstract
We study via Monte Carlo simulation a generalisation of the so-called vertex interacting
self-avoiding walk (VISAW) model on the square lattice. The configurations are actually
not self-avoiding walks but rather restricted self-avoiding trails (bond avoiding paths)
which may visit a site of the lattice twice provided the path does not cross itself: to
distinguish this subset of trails we shall call these configurations grooves. Three
distinct interactions are added to the configurations: firstly the VISAW interaction,
which is associated with doubly visited sites, secondly a nearest neighbour interaction
in the same fashion as the canonical interacting self-avoiding walk (ISAW) and thirdly,
a stiffness energy to enhance or decrease the probability of bends in the configuration.
In addition to the normal high temperature phase we find three low temperature phases:
(i) the usual amorphous liquid drop-like 'globular' phase, (ii) an anisotropic 'β-sheet'
phase with dominant configurations consisting of aligned long straight segments, which
has been found in semi-flexible nearest neighbour ISAW models, and (iii) a maximally dense
phase, where the all sites of the path are associated with doubly visited sites (except
those of the boundary of the configuration), previously observed in interacting self-avoiding trails.
We construct a phase diagram using the fluctuations of the energy parameters and three order parameters.
The β-sheet and maximally dense phases do not seem to meet in the phase space and are always separated
by either the extended or globular phases. We focus attention on the transition between the extended
and maximally dense phases, as that is the transition in the original VISAW model. We find that for
the path lengths considered there is a range of parameters where the transition is first order
and it is otherwise continuous.
Eduardo Dagrosa, and Aleksander L. Owczarek and Thomas Prellberg
Writhe induced phase transition in unknotted self-avoiding polygons
J. Stat. Mech.: Th. and Exp. 50 093206 (20pp) (2017)
Link | PDF file |
arXiv |
Show/Hide Abstract
Recently it has been argued that weighting the writhe of
unknotted self-avoiding polygons can be related to possible experiments that turn double stranded DNA.
We first solve exactly a directed model and demonstrate that in such a subset of polygons the problem
of weighting their writhe is associated with a phase transition. We then analyse simulations using the
Wang-Landau algorithm to observe scaling in the fluctuations of the writhe that is
compatible with a second-order phase transition in a undirected self-avoiding polygon model.
The transition can be clearly detected when the polygon is stretched with a strong pulling force.
Eduardo Dagrosa, and Aleksander L. Owczarek and Thomas Prellberg
Phase Diagram of Twist Storing Lattice Polymers in Variable Solvent Quality
J. Stat. Mech.: Th. and Exp. 50 103204 (18pp) (2017)
Link | PDF file |
arXiv |
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When double stranded DNA is turned in experiments
it undergoes a transition. We use an interacting self-avoiding walk on a three-dimensional fcc
lattice weighted by writhe to relate to these experiments and treat this problem via simulations.
We provide evidence for the existence of a thermodynamic phase transition induced
by writhe and examine related phase diagrams taking solvent quality and stretching into account.
Aleksander L. Owczarek and Andrew Rechnitzer
Force signature of the unzipping transition for strip confined two-dimensional polymers
J. Phys. A: Math. Theor. 50 484001 (24pp) (2017)
[Artwork from this article appears on the front cover of the journal and the article also appeared in the Journal of Physics
A Highlights of 2017 collection]
Link | PDF file |
Show/Hide Abstract
We find and analyse the exact solution of two friendly walks,
modelling polymers, confined between two parallel walls in a two-dimensional strip (or slit) where the
polymers interact with each other via an attractive contact interaction. In the bulk, where the polymers
are always far from any walls, there is an unzipping transition between phases where the two walks drift
away for low attractive fugacity (high temperatures) and bind together for high attractive fugacities
(low temperatures). Previously this has been used to model the denaturation of DNA.
In a strip the transition is not sharp. However, we demonstrate that there is an abrupt change
in the repulsive force
exerted on the walls of the strip that can be calculated exactly. We suggest that this change
in the force could be exploited to provide an experimental signature of the unzipping transition.
Andrea Bedini, Aleksander L. Owczarek and Thomas Prellberg
The role of three-body interactions in two-dimensional polymer collapse
J. Phys. A: Math. Theor. 49 214001 (16pp) (2016)
Link | PDF file |
arXiv |
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Various interacting lattice path models of polymer collapse in
two dimensions demonstrate different critical behaviours. This difference has been without a clear explanation.
The collapse transition has been variously seen to be in the Duplantier–Saleur θ-point university class
(specific heat cusp), the interacting trail class (specific heat divergence) or even first-order.
Here we study via Monte Carlo simulation a generalisation of the Duplantier–Saleur model on the honeycomb
lattice and also a generalisation of the so-called vertex-interacting self-avoiding walk model
(configurations are actually restricted trails known as grooves) on the triangular lattice.
Crucially for both models we have three- and two-body interactions explicitly and differentially weighted.
We show that both models have similar phase diagrams when considered in these larger two-parameter spaces.
They demonstrate regions for which the collapse transition is first-order for high three-body interactions
and regions where the collapse is second order. We conjecture a higher order multiple critical point
separating these two types of collapse.
It remains to be tested whether the second order lines in both models are in the Duplantier–Saleur θ-point
university class for all values of the parameters.
Rami Tabbara, Aleksander L. Owczarek and Andrew Rechnitzer
An exact solution of three interacting friendly walks in the bulk
J. Phys. A: Math. Theor. 49 154004 (27pp) (2016)
Link | PDF file |
arXiv |
Show/Hide Abstract
We find the exact solution of three interacting friendly
directed walks on the square lattice in the bulk, modelling a system of homopolymers that can undergo
a multiple polymer fusion or zipping transition by introducing two distinct interaction parameters that
differentiate between the zipping of only two or all three walks. We establish functional equations for
the model's corresponding generating function that are subsequently solved exactly by means of the obstinate
kernel method. We then proceed to analyse our model, first considering the case where triple-walk interaction
effects are ignored, finding that our model exhibits two phases which we classify as free and gelated (or zipped)
regions, with the system exhibiting a second-order phase transition. We then analyse
the full model where both interaction parameters are incorporated, presenting the full phase diagram
and highlighting the additional existence of a first-order gelation (zipping) boundary.
Eduardo Dagrosa, Aleksander L. Owczarek and Thomas Prellberg
Writhe-induced knotting in a lattice polymer
J. Phys. A: Math. Theor. 48 065002 (14pp) (2015)
Link | PDF file |
arXiv |
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We consider a simple lattice model of a topological phase
transition in open polymers. To be precise, we study a model of self-avoiding walks on the simple cubic lattice
tethered to a surface and weighted by an appropriately defined writhe. We also consider the effect of pulling the
untethered end of the polymer from the surface. Regardless of the force we find a first-order phase transition which we
argue is a consequence of increased knotting in the lattice polymer, rather than due to other
effects such as the formation of plectonemes.
Jarek Krawczyk, Aleksander L. Owczarek and Thomas Prellberg
Semi-flexible attracting-segment model of three-dimensional polymer collapse
Physica A 431 74–83 (2015)
Link | PDF file |
arXiv |
Show/Hide Abstract
Recently it has been shown that a two-dimensional model
of self-attracting polymers based on attracting segments with the addition of stiffness displays
three phases: a swollen phase, a globular, liquid-like phase, and an anisotropic crystal-like phase.
Here, we consider the attracting segment model in three dimensions with the addition of stiffness.
While we again identify a swollen and two distinct collapsed phases,
we find that both collapsed phases are anisotropic, so that there is no
phase in which the polymer resembles a disordered liquid drop. Moreover all
the phase transitions are first order.
Rami Tabbara, Aleksander L. Owczarek and Andrew Rechnitzer
An exact solution of two friendly interacting directed walks near a sticky wall
J. Phys. A: Math. Theor. 47 015202 (34pp) (2014)
Link | PDF file |
Show/Hide Abstract
We find the exact solution of two interacting friendly directed walks (modelling polymers)
on the square lattice. These walks are confined to the quarter plane by a horizontal attractive surface,
to capture the effects of DNA-denaturation and adsorption. We find the solution to the model's
corresponding generating function by means of the obstinate kernel method. Specifically,
we apply this technique in two different instances to establish partial solutions for two
simplified generating functions of the same underlying model that ignore either surface or
shared contacts. We then subsequently combine our two partial solutions to find the solution
for the full generating function in terms of the two simpler variants. This expression
guides our analysis of the model, where we find the system exhibits four phases, and proceed
to delineate the full phase diagram, showing that all observed phase transitions are second-order.
Heather Lonsdale, Iwan Jensen, John N. Essam and Aleksander L. Owczarek
Analysis of mean cluster size in directed compact percolation near a damp wall
J. Stat. Mech: Th. and Exp. P03004 (20pp) (2014)
Link | PDF file |
arXiv |
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We investigate the behaviour of the mean size of directed compact
percolation clusters near a damp wall in the low-density region, where sites in the bulk are wet (occupied)
with probability p while sites on the wall are wet with probability pw. Methods used to find the exact solution
for the dry case (pw = 0) and the wet case (pw = 1) turn out to be inadequate for the damp case. Instead we use
a series expansion for the pw = 2p case to obtain a second-order inhomogeneous differential equation satisfied
by the mean size, which exhibits a critical exponent γ = 2, in common with the wet wall result. For the more
general case of pw = rp, with r rational, we use a modular arithmetic method for finding ordinary differential
equations (ODEs) and obtain a fourth-order homogeneous
ODE satisfied by the series. The ODE is expressed exactly in terms of r.
We find that in the damp region 0 < r < 2 the critical exponent γdamp = 1,
which is the same as the dry wall result.
Eduardo Dagrosa and Aleksander L. Owczarek
Generalizing Ribbons and the Twist of the Lattice Ribbon
J. Stat. Phys 145 392–417 (2014)
Link | PDF file |
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For a standard or canonical ribbon from differential geometry
the topological White’s theorem connects the linking number, writhe and total twist of the ribbon.
Here we provide an integral expression, analog to the total twist of a canonical ribbon, that connects
linking number and writhe of two curves that do not necessarily form a canonical ribbon. First,
we apply this integral expression to derive an expression for the writhe of a polygonal curve.
Second, but importantly, we revisit the lattice ribbon. Lattice ribbons were introduced some time
ago to enable simulation of physical systems modeled by double stranded polymers. Application of the integral
expression yields an algorithm for determining the twist of the lattice ribbon. An interesting relation
between writhe of the center line of a lattice ribbon and its linking number follows.
Andrea Bedini, Aleksander L. Owczarek and Thomas Prellberg
Lattice polymers with two competing collapse interactions
J. Phys. A: Math. Theor. 47 145002 (10pp) (2014)
Link | PDF file |
arXiv |
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We study a generalized model of self-avoiding trails, c
ontaining two different types of interaction (nearest-neighbour contacts and multiply visited sites),
using computer simulations. This model contains various previously studied models as special cases.
We find that the strong collapse transition induced by multiply-visited sites is a singular point in
the phase diagram and corresponds to a higher order
multi-critical point separating a line of weak second-order transitions from a line of first-order transitions.
Aleksander L. Owczarek and Thomas Prellberg
Pressure exerted by a vesicle on a surface
J. Phys. A: Math. Theor. 47 215001 (9pp) (2014)
Link | PDF file |
arXiv |
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Several recent works have considered the pressure exerted
on a wall by a model polymer. We extend this consideration to vesicles attached to a wall, and hence
include osmotic pressure. We do this by considering a two-dimensional directed model, namely that of a
rea-weighted Dyck paths. Not surprisingly, the pressure exerted by the vesicle on the wall depends on
the osmotic pressure inside, especially its sign. Here, we discuss the scaling of this pressure in the
different regimes, paying particular attention to the crossover between positive and negative osmotic pressure.
In our directed model, there exists an underlying Airy function scaling form, from which we extract the
dependence of the bulk pressure on small osmotic pressures.
Thomas Wong, Aleksander L. Owczarek and Andrew Rechnitzer
Confining multiple polymers between sticky walls: a directed walk model of two polymers
J. Phys. A: Math. Theor. 47 415002 (37pp) (2014)
Link | PDF file |
arXiv |
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We study a model of two polymers confined to a slit with sticky walls.
More precisely, we find and analyse the exact solution of two directed friendly walks in such a geometry on the
square lattice. We compare the infinite slit limit, in which the length of the polymer (thermodynamic limit) is
taken to infinity before the width of the slit is considered to become large, to the opposite situation where the
order of the limits are swapped, known as the half-plane limit when one polymer is modelled. In contrast with the
single polymer system we find that the half-plane and infinite slit limits coincide. We understand this result in
part due to the tethering of polymers on both walls of the slit. We also analyse the entropic force exerted by the
polymers on the walls of the slit. Again the results differ significantly from single polymer models. In a single
polymer system both attractive and repulsive regimes were seen, whereas in our two walk model only repulsive forces
are observed. We do, however,
see that the range of the repulsive force is dependent on the parameter values. This variation can be explained by
the adsorption of the walks on opposite walls of the slit.
Andrea Bedini, Aleksander L. Owczarek and Thomas Prellberg
Semi-flexible interacting self-avoiding trails on the square lattice
Physica A 392 1602–10 (2013)
Link | PDF file |
arXiv |
Show/Hide Abstract
Self-avoiding walks self-interacting via nearest neighbours (ISAW)
and self-avoiding trails interacting via multiply-visited sites (ISAT) are two models of the polymer collapse
transition of a polymer in a dilute solution. On the square lattice it has been established numerically that
the collapse transition of each model lies in a different universality class.
It has been shown that by adding stiffness to the ISAW model a second low temperature phase eventuates and a more
complicated phase diagram ensues with three types of transition that meet at a multi-critical point. For large enough
stiffness the collapse transition becomes first order. Interestingly, a phase diagram of a similar structure has been
seen to occur in an extended ISAT model on the triangular lattice without stiffness. It is therefore of interest to see
the effect of adding stiffness to the ISAT model.
We have studied by computer simulation a generalised model of self-interacting self-avoiding trails on the square
lattice with a stiffness parameter added. Intriguingly, we find that stiffness does not change the order of the
collapse transition for ISAT on the square lattice for a very wide range of stiffness weights. While at the
lengths considered there are clear bimodal distributions for very large stiffness, our numerical evidence strongly
suggests that these are simply finite-size effects associated with a crossover to a first-order phase transition at infinite stiffness.
Andrea Bedini, Aleksander L. Owczarek and Thomas Prellberg
Self-avoiding trails with nearest neighbour interactions on the square lattice
J. Phys. A: Math. Theor. 46 085001 (13pp) (2013)
Link | PDF file |
arXiv |
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Self-avoiding walks and self-avoiding trails, two models of a polymer coil in dilute solution,
have been shown to be governed by the same universality class. On the other hand, self-avoiding
walks interacting via nearest-neighbour contacts (ISAW) and self-avoiding trails interacting via
multiply visited sites (ISAT) are two models of the coil-globule, or collapse transition of a
polymer in dilute solution. On the square lattice it has been established numerically that the
collapse transition of each model lies in a different universality class. The models differ in
two substantial ways. They differ in the types of subsets of random walk configurations utilized
(site self-avoidance versus bond self-avoidance) and in the type of attractive interaction. It is
therefore of some interest to consider self-avoiding trails interacting via nearest-neighbour attraction
(INNSAT) in order to ascertain the source of the difference in the collapse universality class.
Using the flatPERM algorithm, we have performed computer simulations of this model. We present
numerical evidence that the singularity in the free energy of INNSAT at the collapse transition
has a similar
exponent to that of the ISAW model rather than the ISAT model. This would indicate that the type
of interaction used in ISAW and ISAT is the source of the difference in the universality class.
Andrea Bedini, Aleksander L. Owczarek and Thomas Prellberg
Weighting of topologically different interactions in a model of two-dimensional polymer collapse
Phys. Rev. E 87 012142 (8pp) (2013)
Link | PDF file |
arXiv |
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We study by computer simulation a recently
introduced generalized model of self-interacting self-avoiding trails on the square lattice
that distinguishes two topologically different types of self-interaction: namely, crossings
where the trail passes across itself and collisions where the lattice path visits the same
site without crossing. This model generalizes the canonical interacting self-avoiding trail
model of polymer collapse, which has a strongly divergent specific heat at its transition point.
We confirm the recent prediction that the asymmetry does not affect the universality class
for a range of asymmetry. Certainly, where the weighting of collisions outweighs that of
crossings this is well supported numerically.
When crossings are weighted heavily relative to collisions, the collapse transition reverts
to the canonical 𝜃-point-like behavior found in interacting self-avoiding walks.
Andrea Bedini, Aleksander L. Owczarek and Thomas Prellberg
Numerical simulation of a lattice polymer model at its integrable point
J. Phys. A: Math. Theor. 46 265003 (9pp) (2013)
Link | PDF file |
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We revisit an integrable lattice model of polymer
collapse using numerical simulations. This model was first studied by Blöte and Nienhuis
(1989 J. Phys. A: Math. Gen. 22 1415) and it describes polymers with some attraction,
providing thus a model for the polymer collapse transition. At a particular set of Boltzmann
weights the model is integrable and the exponents ν = 12/23 ≈ 0.522 and γ = 53/46 ≈ 1.152 have
been computed via identification of the scaling dimensions xt = 1/12 and xh = −5/48.
We directly investigate the polymer scaling exponents via Monte Carlo simulations using the
pruned-enriched Rosenbluth method algorithm. By simulating this polymer model for walks up to
length 4096 we find ν = 0.576(6) and γ = 1.045(5), which are clearly different from the predicted values.
Our estimate for the exponent ν is compatible with the known θ-point value of 4/7 and in agreement
with very recent numerical evaluation by Foster and Pinettes (2012 J. Phys. A: Math. Theor. 45 505003).
Andrea Bedini, Aleksander L. Owczarek and Thomas Prellberg
Anomalous critical behaviour in the polymer collapse transition of three-dimensional lattice trails
Phys. Rev. E 86 011123 (10pp) (2012)
Link | PDF file |
arXiv |
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Trails (bond-avoiding walks) provide an alternative lattice model
of polymers to self-avoiding walks, and adding self-interaction at multiply visited sites gives a
model of polymer collapse. Recently a two-dimensional model (triangular lattice) where doubly and
triply visited sites are given different weights was shown to display a rich phase diagram with
first- and second-order collapse separated by a multicritical point. A kinetic growth process of
trails (KGTs) was conjectured to map precisely to this multicritical point. Two types of
low-temperature phases, a globule phase and a maximally dense phase, were encountered.
Here we investigate the collapse properties of a similar extended model of interacting
lattice trails on the simple cubic lattice with separate weights for doubly and triply visited sites.
Again we find first- and second-order collapse transitions dependent on the relative sizes of the
doubly and triply visited energies. However, we find no evidence of a low-temperature maximally
dense phase with only the globular phase in existence. Intriguingly, when the ratio of the energies
is precisely that which separates the first-order from the second-order regions anomalous
finite-size scaling appears. At the finite-size location of the rounded transition clear
evidence exists for a first-order transition that persists in the thermodynamic limit.
This location moves as the length increases, with its limit apparently at the point that maps to a KGT.
However, if one fixes the temperature to sit at exactly this KGT point, then only a
critical point can be deduced from the data. The resolution of this apparent contradiction lies in the
breaking of crossover scaling and the difference in the shift and transition width (crossover) exponents.
Aleksander L. Owczarek and Thomas Prellberg
Enumeration of area-weighted Dyck path with restricted height Aust. J. Comb. 54 13–8 (2012)
Link | PDF file |
arXiv |
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We derive explicit expressions for q-orthogonal polynomials arising in the
enumeration of area-weighted Dyck paths with restricted height.
Aleksander L. Owczarek and Thomas Prellberg
Exact solution of a model of a vesicle attached to a wall subject to mechanical deformation
J. Phys. A: Math. Theor. 45 395001 (10pp) (2012)
Link | PDF file |
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Area-weighted Dyck-paths are a two-dimensional model
for vesicles attached to a wall. We model the mechanical response of a vesicle to a
pulling force by extending this model. We obtain an exact solution using two different approaches,
leading to a q-deformation of an algebraic functional equation, and a q-deformation of a
linear functional equation with a catalytic variable, respectively. While the
non-deformed linear functional equation is solved by substitution of
special values of the catalytic variable (the so-called kernel method),
the q-deformed case is solved by iterative substitution of the catalytic variable.
Our model shows a non-trivial phase transition when a
pulling force is applied. As soon as the area is weighted with non-unity weight,
this transition vanishes.
Aleksander L. Owczarek, Andrew Rechnitzer and Thomas Wong
Exact solution of two friendly walks above a sticky wall with single and double interactions
J. Phys. A: Math. Theor. 45 425003 (23pp) (2012)
Link | PDF file |
Show/Hide Abstract
We find, and analyse, the exact
solution of two friendly directed walks, modelling polymers, which interact with
a wall via contact interactions. We specifically consider two walks that begin and
end together so as to imitate a polygon. We examine a general model in which a separate
interaction parameter is assigned to configurations where both polymers touch the wall simultaneously,
and investigate the effect this parameter has on the integrability of the problem.
We find an exact solution of the generating function of the model,
and provide a full analysis of the phase diagram that admits three phases
ith one first-order and two second-order transition lines between these phases.
We argue that one physically realizable model would see two phase transitions as
the temperature is lowered.
Rami Tabbara and Aleksander L. Owczarek
Pulling a polymer with anisotropic stiffness near a sticky wall
J. Phys. A. 45 435002 (40pp) (2012)
Link | PDF file |
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We solve exactly a two-dimensional partially directed walk model
of a semi-flexible polymer that has one end tethered to a sticky wall, while a pulling force away
from the adsorbing surface acts on the free end of the walk. This model generalizes a number of
previously considered adsorption models by incorporating individual horizontal and vertical stiffness effects,
in competition with a variable pulling angle. A solution to the corresponding generating function
is found by means of the kernel method. While the phases and related phase transitions are similar
in nature to those found previously the analysis of the model in terms of its physical variables
highlights various novel structures in the shapes of the phase diagrams and related behaviour of the polymer.
We review the results of previously considered sub-cases, augmenting these findings to include analysis with
respect to the model's physical variables—namely, temperature, pulling force, pulling angle away from the surface,
stiffness strength and the ratio of vertical to horizontal stiffness potentials, with our subsequent analysis for the
general model focusing on the effect that stiffness has on this pulling angle range. In analysing the model with
stiffness we also pay special attention to the case where only vertical stiffness is included.
The physical analysis of this case reveals behaviour more closely resembling that of an upward pulling force
acting on a polymer than it does of a model where horizontal stiffness acts. The stiffness–temperature phase diagram
exhibits re-entrance for low temperatures, previously only seen for three-dimensional or co-polymer models.
For the most general model we delineate the shift in the physical behaviour as we change the ratio of vertical to horizontal stiffness between the horizontal-only
and the vertical-only stiffness regimes. We find that a number of distinct physical characteristics
will only be observed for a model where the vertical stiffness dominates the horizontal stiffness.
Heather Lonsdale and Aleksander L. Owczarek
Directed compact percolation near a damp wall with biased growth
J. Stat. Mech. P11001 (18pp) (2012)
Link | PDF file |
Show/Hide Abstract
he model of directed compact percolation near a damp wall is
generalized to allow for a bias in the growth of a cluster, either towards or away from the wall. T
he percolation probability for clusters beginning with seed width m, any distance from the wall,
is derived exactly by solving the associated recurrences. It is found that the general biased case
near a damp wall leads to a
critical exponent β = 1, in line with the dry biased case, which differs from the unbiased damp/dry exponent β = 2.
John N. Essam, Heather Lonsdale and Aleksander L. Owczarek
Mean length of finite clusters in directed compact percolation near a damp wall
J. Stat. Phys 145 639–46 (2011)
Link | PDF file |
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The mean length of finite clusters is derived exactly
for the case of directed compact percolation near a damp wall. We find
that the result involves elliptic integrals and exhibits similar critical behaviour to the dry wall case.
Heather Lonsdale, John N. Essam and Aleksander L. Owczarek
Directed compact percolation near a damp wall: mean length and mean number of wall contacts
J. Phys. A: Math. Theor. 44 505003 (26pp) (2011)
Link | PDF file |
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Key aspects of the cluster distribution in the case of
directed, compact percolation near a damp wall are derived as functions of the bulk occupation probability
p and the wall occupation probability pw. The mean length of finite clusters and mean number of contacts with
the wall are derived exactly, and we find that both results involve elliptic integrals and further multiple sum
functions of two variables. Despite the added complication of
these multiple sum functions, our analysis shows that the critical behaviour is similar to the dry wall
case where these functions do not appear. We derive the critical amplitudes as a function of pw.
Jarek Krawczyk, Aleksander L. Owczarek and Thomas Prellberg
Semi-flexible attracting-segment model of two-dimensional polymer collapse
Physica A 369 1619–24 (2010)
Link | PDF file |
Show/Hide Abstract
Recently it has been shown that a two-dimensional model of self-attracting
polymers based on attracting segments displays two phase transitions, a
-like collapse between swollen polymers and a globular state and another between the globular state and a polymer crystal.
On the other hand, the canonical model based on attracting monomers on lattice sites displays only one: the standard tricritical
collapse transition. Here we consider the attracting segment model with the addition of stiffness
and show that it displays the same phases as the canonical model.
Richard Brak, Gary K Iliev, Aleksander L. Owczarek and Stu G. Whittington
Exact solution of a three-dimensional lattice polymer confined in a slab with sticky walls
J. Phys. A.: Math. Theor. 43 135001 (12pp) (2010)
Link | PDF file |
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We present the exact solution of a three-dimensional lattice model
of a polymer confined between two sticky walls, that is within a slab. We demonstrate that the model behaves
in a similar way to its two-dimensional analogues and agrees with Monte Carlo evidence based upon
simulations of self-avoiding walks in slabs. The model on which we focus is a variant of the
partially directed walk model on the cubic lattice.
We consider both the phase diagram of relatively long polymers in a macroscopic slab
and the effective force of the polymer on the walls of the slab.
Aleksander L. Owczarek
Effect of stiffness on the pulling of an adsorbing polymer from a wall: Exact Solution of a Partially Directed Walk model
J. Phys. A.: Math. Theor. 43 225002 (16pp) (2010)
Link | PDF file |
Show/Hide Abstract
Recently the effect of stiffness, or semi-flexibility,
on the adsorption and also the collapse phase transitions of isolated polymers has been explored
via the exact solutions of partially directed walk models. Here we consider its effect on the
stretching transition mediated
by the application of a force to one end of the polymer when the other end is attached to an adsorbing wall.
Aleksander L. Owczarek and Thomas Prellberg
Exact Solution of the Discrete (1+1)–dimensional RSOS Model in a Slit with Field and Wall Interaction
J. Phys. A.: Math. Theor. 43 375004 (10pp) (2010)
Link | PDF file |
arXiv |
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We present the solution of a linear restricted solid-on-solid (RSOS) model confined to a slit.
We include a field-like energy, which equivalently weights the area under the interface,
and also include independent interaction terms with both walls. This model can also be mapped
to a lattice polymer model of Motzkin paths in a slit interacting with both walls including
an osmotic pressure. This work generalizes the previous work on the RSOS model in the half-plane
which has a solution that was shown recently to exhibit a novel mathematical structure involving
basic hypergeometric functions 3ϕ2. Because of the mathematical relationship between the half-plane
and slit this work hence effectively explores the underlying q-orthogonal polynomial structure to that solution.
It also generalizes two other recent works: one on Dyck paths weighted with an osmotic pressure in a
slit and another concerning Motzkin paths without an osmotic pressure term in a slit.
Aleksander L. Owczarek and Thomas Prellberg
A simple model of vesicle drop in a confined geometry
J. Stat. Mech.: Theor. Exp. P08015 (13pp) (2010)
Link | PDF file |
arXiv |
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We present the exact solution of a two-dimensional directed walk model
of a drop, or half-vesicle, confined between two walls, and attached to one wall. This model is also a generalization
of a polymer model of steric stabilization recently investigated. We explore the competition between a sticky potential
on the two walls and the effect of a pressure-like term in the system. We show that a
negative pressure ensures the drop/polymer is unaffected by confinement when the walls are a macroscopic distance apart.
Jason Doukas, Aleksander L. Owczarek and Thomas Prellberg
Identification of a polymer growth process with an equilibrium multi-critical collapse phase transition: the meeting point of swollen, collapsed and crystalline polymers
Phys. Rev. E 82 031103 (12pp) (2010)
Link | PDF file |
arXiv |
Show/Hide Abstract
We have investigated a polymer growth process on the triangular lattice
where the configurations produced are self-avoiding trails. We show that the scaling behavior of this process is
similar to the analogous process on the square lattice. However, while the square lattice process maps to the
collapse transition of the canonical interacting self-avoiding trail (ISAT) model on that lattice,
the process on the triangular lattice model does not map to the canonical equilibrium model. On the other hand,
we show that the collapse transition of the canonical ISAT model on the triangular lattice behaves in a way
reminiscent of the 𝜃 point of the interacting self-avoiding walk (ISAW) model, which is the standard model of polymer collapse.
This implies an unusual lattice dependency of the ISAT collapse transition in two dimensions. By studying an extended ISAT model,
we demonstrate that the growth process maps to a multicritical point in a larger parameter space.
In this extended parameter space the collapse phase transition may be either 𝜃-point-like (second order)
or first order, and these two are separated by a multicritical point. It is this multicritical point to which the growth process maps.
Furthermore, we provide evidence that in addition to the high-temperature gaslike swollen polymer phase (coil)
and the low-temperature liquid-drop-like collapse phase (globule) there is also a maximally dense crystal-like phase (crystal)
at low temperatures dependent on the parameter values. The multicritical point is the meeting point of these three phases.
Our hypothesized phase diagram resolves the mystery of the seemingly differing behaviors of the ISAW and ISAT models
in two dimensions as well as the behavior of the trail growth process.
Jarek Krawczyk, Aleksander L. Owczarek and Thomas Prellberg
Semi-flexible hydrogen-bonded and non-hydrogen bonded lattice polymers
Physica A 388 104–12 (2009)
Link | PDF file |
Show/Hide Abstract
We investigate the addition of stiffness to the lattice model of
hydrogen-bonded polymers in two and three dimensions. We find that, in contrast to polymers that interact
via a homogeneous short-range interaction, the collapse transition is unchanged by any amount of stiffness:
this supports the physical argument that hydrogen bonding already introduces an effective stiffness.
Contrary to possible physical arguments, favouring bends in the polymer does not return the model’s behaviour
to that comparable with the semi-flexible homogeneous interaction model, where the canonical
-point occurs for a range of parameter values. In fact, for sufficiently large bending energies the
crystal phase disappears altogether, and no phase transition of any type occurs. We also compare the
order-disorder transition from the globule phase to crystalline phase in the semi-flexible
homogeneous interaction model to that for the fully-flexible hybrid model with both hydrogen and non-hydrogen
like interactions. We show that these phase transitions are of the same type and are a novel polymer critical phenomena
in two dimensions. That is, it is confirmed that in two dimensions this transition is second-order, unlike in
three dimensions where it is known to be first order. We also estimate the crossover exponent in two dimensions
and show that there is a divergent specific heat, finding
or equivalently
. This is therefore different from the
transition, for which
.
Richard Brak, Patrick Dyke, Jennifer Lee, Aleksander L. Owczarek, Thomas Prellberg, Andrew Rechnitzer and Stuart G. Whittington
A self-interacting partially directed walk subject to a force
J. Phys. A.: Math. Theor. 42 085001(30pp) (2009)
Link | PDF file |
arXiv |
Show/Hide Abstract
We consider a directed walk model of a homopolymer (in two dimensions)
which is self-interacting and can undergo a collapse transition, subject to an applied tensile force. We review and
interpret all the results already in the literature concerning the case where this force is in the preferred direction
of the walk. We consider the force extension curves at different temperatures as well as the critical-force temperature curve.
We demonstrate that this model can be analysed rigorously for all key quantities of interest even when
there may not be explicit expressions for these quantities available. We show which of the techniques available
can be extended to the full model, where the force has components in the preferred direction and the direction
perpendicular to this. Whilst the solution of the generating function is available, its analysis is far more
complicated and not all the rigorous techniques are available. However, many results can be extracted including
the location of the critical point which gives the general critical-force temperature curve.
Lastly, we generalize the model to a three-dimensional analogue and show that several key properties
can be analysed if the force is restricted to the plane of preferred directions.
Heather Lonsdale, Richard Brak, John N. Essam, Aleksander L. Owczarek and Andrew Rechnitzer
On directed compact percolation near a damp wall
J. Phys. A.: Math. Theor. 42 125001(26pp) (2009)
[This article was highlighted by the journal as one of the most highly rated by the referees.]
Link | PDF file |
Show/Hide Abstract
A percolation probability for directed, compact percolation near a damp wall,
which interpolates between the previously examined cases, is derived exactly. We find that the critical exponent β = 2
in common with the dry wall, rather than the value previously found in the wet wall and bulk cases.
The solution is found via a mapping to a particular model of directed walks. We evaluate the exact generating
function for this walk model which is also related to the ASEP model of traffic flow.
We compare the underlying mathematical structure of the various cases previously considered
and this one by reviewing the common framework of solution via the mapping to different directed walk models.
Jarek Krawczyk, Iwan Jensen, Aleksander L. Owczarek and Sanjay Kumar
Pulling self-interacting polymers in two-dimensions
Phys. Rev. E 031912:(9pp) (2009)
Link | PDF file |
arXiv |
Show/Hide Abstract
We investigate a two-dimensional problem of an isolated
self-interacting end-grafted polymer, pulled by one end. In the thermodynamic limit,
we find that the model has only two different phases, namely a collapsed phase and a stretched phase.
We show that the phase diagram obtained by Kumar et al. [Phys. Rev. Lett. 98, 128101 (2007)] for small systems,
where differences between various statistical ensembles play an important role,
differs from the phase diagram obtained here in the thermodynamic limit.
Aleksander L. Owczarek
Exact Solution of Semi-Flexible Partially Directed Walks at an Adsorbing Wall
J. Stat. Mech.: Theor. Exp. P11002:1-15 (2009)
Link | PDF file |
Show/Hide Abstract
Recently it was shown that the introduction of stiffness into the
model of self-interacting partially directed walks modifies the polymer collapse transition seen from a
second-order to a first-order one. Here we consider the effect of stiffness on the adsorption transition.
We provide the exact generating function for non-interacting semi-flexible partially directed walks and
analyse the solution in detail. We demonstrate that stiffness does not change the order of the adsorption transition,
in contrast to its effect on collapse.
Aleksander L. Owczarek and Thomas Prellberg
Exact Solution of the Discrete (1+1)–dimensional RSOS Model with Field and Surface Interactions
J. Phys. A.: Math. Theor. 42 495003 (9pp) (2009)
Link | PDF file |
arXiv |
Show/Hide Abstract
We present the solution of a linear restricted solid-on-solid (RSOS) model
in a field. Aside from the origins of this model in the context of describing the phase boundary in a magnet,
interest also comes from more recent work on the steady state of non-equilibrium models of molecular motors.
While similar to a previously solved (non-restricted) SOS model in its physical behaviour,
mathematically the solution is more complex.
Involving basic hypergeometric functions 3ϕ2,
it introduces a new form of solution to the lexicon of directed lattice path generating functions.
Aleksander L. Owczarek
Exact results for a Directed Polymer Model Related to Quantum Entanglement in Far from Equilibrium Stationary States
J. Stat. Mech.: Theor. Exp. P12004:1-14 (2009)
Link | PDF file |
Show/Hide Abstract
A directed polymer model has been used by Alcaraz et al to
reflect properties related to models of quantum entanglement in far from equilibrium stationary states.
Here we calculate exactly one property related to entanglement: the average height of the polymer from the surface.
In doing so we extend a well known method of exact solution for directed polymer models.
Of interest from the polymer viewpoint is that
the average height is asymptotically independent of the stickiness in the desorbed phase.
Aleksander L. Owczarek, Richard Brak and Andrew Rechnitzer
Self-avoiding walks in slits and slabs with interactive walls
J. Math. Chem. 45 113–28 (2009)
Link | PDF file |
Show/Hide Abstract
Self-avoiding walk models of a polymer confined between
two parallel attractive walls in two and three dimensions (slits and slabs, respectively) have
recently had a revival of interest. They were first studied as simple models of steric stabilisation
and sensitised flocculation in colloids. The revival has been catalysed by new exact solution techniques,
that have allowed the solution of directed walk models in two dimensions in full generality, and by
new Monte Carlo techniques that have allowed the simulation of the full parameter space in the
three-dimensional slab model. Additionally, rigorous techniques applied to the slab problem
have also yielded new results. The contributions to the study of this problem that have
been recently added include a novel phase diagram for the “infinite-slab” (when the walls
are a macroscopic distance apart but both walls may
still “see” the polymer) the delineation of the repulsive and attractive regimes of the parameter space,
and a conjectured scaling theory for the problem in general dimensions.
Richard Brak, Aleksander L. Owczarek and Andrew Rechnitzer
Exact solutions of some lattice polymer models
J. Math. Chem. 45 39–57 (2009)
Link | PDF file |
Show/Hide Abstract
We consider directed path models of a selection of polymer and vesicle problems.
Each model is used to illustrate an important method of solving lattice path enumeration problems.
In particular, the Temperley method is used for the polymer collapse problem.
The ZL method is used to solve the semi-continuous vesicle model.
The Constant Term method is used to solve a set of partial difference equations for the polymer adsorption problem.
The Kernel method is used to solve the functional equation that arises in the polymer force problem.
Finally, the Transfer Matrix method is used to solve a problem in colloid dispersions.
All these methods are combinatorially similar as they all construct equations by
considering the action of adding an additional column to the set of objects.
Aleksander L. Owczarek, Thomas Prellberg and Andrew Rechnitzer
Finite-size scaling functions for directed polymers confined between attracting walls
J. Phys. A.: Math. Theor. 41 035002:1-16 (2008)
Link | PDF file |
arXiv |
Show/Hide Abstract
The exact solution of directed self-avoiding walks
confined to a slit of finite width and interacting with the walls of the slit via an attractive
potential has been recently calculated. The walks can be considered to model the polymer-induced
steric stabilization and sensitized flocculation of colloidal dispersions. The large-width asymptotics
led to a phase diagram different to that of a polymer attached to, and attracted to, a single wall.
The question that arises is: Can one interpolate between the single wall and two wall cases?
In this paper, we calculate the exact scaling functions for the partition function by
considering the two variable asymptotics of the partition function for simultaneous
large length and large width. Consequently, we find the scaling functions for the
force induced by the polymer on the walls. We find that these scaling functions are
given by elliptic ϑ functions.
In some parts of the phase diagram there is more a complex crossover between the
single wall and two wall cases and we elucidate how this happens.
Aleksander L. Owczarek and Thomas Prellberg
Scaling of the Atmosphere of Self-Avoiding Walks
J. Phys. A.: Math. Theor. 41 375004 (6pp) (2008)
Link | PDF file |
arXiv |
Show/Hide Abstract
The number of free sites next to the end of a self-avoiding walk
is known as the atmosphere of the walk. The average atmosphere can be related to the number of configurations.
Here we study the distribution of atmospheres as a function of length and how the number of walks of fixed atmosphere scale.
Certain bounds on these numbers can be proved. We use Monte Carlo estimates to verify our conjectures in two dimensions.
Of particular interest are walks that have zero atmosphere, which are known as trapped.
We demonstrate that these walks scale in the same way as the full set of self-avoiding walks,
barring an overall constant factor.
Yao-Ban Chan, Aleksander L. Owczarek, Andrew Rechnitzer and Gordon Slade
Mean unknotting times of random knots and embeddings
J. Stat. Mech.: Theor. Exp. P05004:1-16 (2007)
Link | PDF file |
Show/Hide Abstract
We study mean unknotting times of knots and knot embeddings by crossing reversals,
in a problem motivated by DNA entanglement. Using self-avoiding polygons (SAPs) and self-avoiding polygon trails (SAPTs)
we prove that the mean unknotting time grows exponentially in the length of the SAPT and at least exponentially with the
length of the SAP. The proof uses Kesten's pattern theorem, together with results for mean first-passage times
in the two-parameter Ehrenfest urn model. We use the pivot algorithm to generate random SAPTs of up to 3000 steps
and calculate the corresponding unknotting times, and find that the mean unknotting time grows very slowly,
even at moderate lengths.
Our methods are quite general—for example, the lower bound on the mean unknotting time applies also to Gaussian random polygons.
Richard Martin, Enzo Orlandini E, Aleksander L. Owczarek, Andrew Rechnitzer and Stuart G. Whittington
Exact enumeration and Monte Carlo results for self-avoiding walks in a slab
J. Phys. A.: Math. Theor. 40 7509–21(2007)
Link | PDF file |
Show/Hide Abstract
We analyse exact enumeration data and Monte Carlo simulation results
for a self-avoiding walk model of a polymer confined between two parallel attractive walls (plates). We use the
exact enumeration data to establish the regions where the polymer exerts an effective attractive force between
the plates and where the polymer exerts an effective repulsive force by estimating the boundary (zero-force)
curve. While the phase boundaries of the phase diagram have previously been conjectured we delineate this
further by establishing the order of the phase transitions for the so-called infinite slab (that is,
when the plates are a macroscopic distance apart). We conclude that the adsorption transitions associated
with either plate are similar in nature to the half-space situation even when a polymer is attached to the
opposite wall. The transition between the two adsorbed phases is established as first order. Importantly,
we conjecture a scaling theory valid in the desorbed and critically
adsorbed regions of the phase diagram and demonstrate the consistency of the Monte Carlo data with these
hypotheses by estimating the corresponding scaling functions.
Aleksander L. Owczarek, Andrew Rechnitzer, Jarek Krawczyk and Thomas Prellberg
On the location of the surface-attached globule phase in collapsing polymers
J. Phys. A: Math. Theor. 40 13257–67(2007)
Link | PDF file |
arXiv |
Show/Hide Abstract
We investigate the existence and location of the
surface phase known as the 'surface-attached globule' (SAG) conjectured previously to exist
in lattice models of three-dimensional polymers when they are attached to a wall that has a
short-range potential. The bulk phase, where the attractive intra-polymer interactions are s
trong enough to cause a collapse of the polymer into a liquid-like globule and the wall either
has weak attractive or repulsive interactions, is usually denoted desorbed-collapsed or DC.
Recently, this DC phase was conjectured to harbour two surface phases separated by a boundary
where the bulk free energy is analytic while the surface free energy is singular. The surface
phase for more attractive values of the wall interaction is the SAG phase. We discuss in more
detail the properties of this proposed surface phase and provide Monte Carlo evidence for
self-avoiding walks up to a length 256 that this surface phase most likely does exist.
Importantly, we discuss alternatives for the surface phase boundary.
In particular, we conclude that this boundary may lie along the zero wall interaction line
and the bulk phase boundaries rather than any new phase boundary curve.
Jarek Krawczyk, Aleksander L. Owczarek and Thomas Prellberg
The competition of hydrogen-like and isotropic interactions on polymer collapse
J. Stat. Mech.: Theor. Exp. P09016:1-15 (2007)
Link | PDF file |
arXiv |
Show/Hide Abstract
We investigate a lattice model of polymers where the nearest
neighbour monomer–monomer interaction strengths differ according to whether the local configurations have
so-called 'hydrogen-like' formations or not. If the interaction strengths are all the same then the classical
θ-point collapse transition occurs on lowering the temperature, and the polymer enters the isotropic liquid
drop phase known as the collapsed globule. On the other hand, strongly favouring the hydrogen-like interactions
gives rise to an anisotropic folded (solid-like) phase on lowering the temperature. We use Monte Carlo
simulations up to a length of 256 to map out the phase diagram in the plane of parameters and determine
the order of the associated phase transitions. We discuss the connections to semi-flexible polymers and
other polymer models. Importantly, we demonstrate that for a range of energy parameters, two phase transitions
occur on lowering the temperature, the second being a transition from the globule state to the crystal state. W
e argue from our data that this
globule-to-crystal transition is continuous in two dimensions in accord with field-theory arguments concerning
Hamiltonian walks, but is first order in three dimensions.
Jarek Krawczyk, Aleksander L. Owczarek, Thomas Prellberg and Andrew Rechnitzer
Lattice model for parallel and orthogonal sheets using hydrogenlike bonding
[Selected for the Virtual Journal of Biological Physics Vol. 14, 2007]
Phys. Rev. E 76 051904:1-5 (2007)
Link | PDF file |
Show/Hide Abstract
We present results for a lattice model of polymers
where the type of 𝛽 sheet formation can be controlled by different types of hydrogen bonds depending
on the relative orientation of close segments of the polymer. Tuning these different interaction strengths
leads to low-temperature structures with different types of orientational order. We perform simulations of
this model and so present the phase diagram,
ascertaining the nature of the phases and the order of the transitions between these phases.
Aleksander L. Owczarek and Thomas Prellberg
Exact solution of semi-flexible and super-flexible interacting partially directed walks
J. Stat. Mech.: Theor. Exp. P11010:1-14 (2007)
Link | PDF file |
arXiv |
Show/Hide Abstract
We provide the exact generating function for semi-flexible and
super-flexible interacting partially directed walks and also analyse the solution in detail.
We demonstrate that while fully flexible walks have a collapse transition that is second order
and obeys tricritical scaling, once positive stiffness is introduced the collapse transition becomes first order.
This confirms a recent conjecture based on numerical results. We note that the addition of a force along the
line of the directness of the walk, in either case, does not affect the order of the transition.
In the opposite case where stiffness is discouraged by the energy potential introduced, which we denote the
super-flexible case, the transition also changes, though more subtly,
with the crossover exponent remaining unmoved from the neutral case but the entropic exponents changing.
Jarek Krawczyk, Aleksander L. Owczarek, Thomas Prellberg and Andrew Rechnitzer
On a Type of Self-Avoiding Random Walk with Multiple Site Weightings and Restrictions
[Selected for the Virtual Journal of Biological Physics Vol. 12, 2006]
Phys. Rev. Lett 96 2406031(1-4) (2006)
Link | PDF file |
Show/Hide Abstract
We introduce a new class of models for polymer collapse,
given by random walks on regular lattices which are weighted according to multiple site visits.
A Boltzmann weight 𝜔𝑙 is assigned to each (𝑙 +1)-fold visited lattice site, and self-avoidance is
incorporated by restricting to a maximal number 𝐾 of visits to any site via setting 𝜔𝑙 =0 for 𝑙 ≥𝐾.
In this Letter we study this model on the square and simple cubic lattices for the case 𝐾 =3.
Moreover, we consider a variant of this model, in which we forbid immediate self-reversal of the random walk.
We perform simulations for random walks up to 𝑛 =1024 steps using FlatPERM,
a flat histogram stochastic growth algorithm.
find evidence that the existence of a collapse transition
depends sensitively on the details of the model and has an unexpected dependence on dimension.
Aleksander L. Owczarek and Thomas Prellberg
Collapse transition of self-avoiding trails on the square lattice
Physica A 373 433–8 (2006)
Link | PDF file |
arXiv |
Show/Hide Abstract
The collapse transition of an isolated polymer has been modelled by many different approaches,
including lattice models based on self-avoiding walks and self-avoiding trails.
In two dimensions, previous simulations of kinetic growth trails, which map to a particular
temperature of interacting self-avoiding trails, showed markedly different behaviour for what was
argued to be the collapse transition than that which has been verified for models based of self-avoiding walks.
On the other hand, it has been argued that kinetic growth trails represent a special simulation that does
not give the correct picture of the standard equilibrium model. In this work we simulate the standard equilibrium
interacting self-avoiding trail model on the square lattice up to lengths over 2,000,000 steps and show that the
results of the kinetic growth simulations are, in fact, entirely in accord with standard simulations of the
temperature dependent model. In this way we verify that the collapse transition of interacting self-avoiding
walks and trails are indeed in different universality classes in two dimensions.
Jarek Krawczyk, Aleksander L. Owczarek, Thomas Prellberg and Andrew Rechnitzer
Pulling absorbing and collapsing polymers from a surface
J. Stat. Mech.: Theor. Exp. P05008(2005)
Link | PDF file |
arXiv |
Show/Hide Abstract
A self-interacting polymer with one end attached to a sticky surface has been
studied by means of a flat-histogram stochastic growth algorithm known as FlatPERM. We examined the four-dimensional
parameter space of the number of monomers (up to 91), self-attraction, surface-attraction and pulling force applied to one end of the polymer.
Using this powerful algorithm the complete parameter space of interactions and pulling force has been considered.
Recently it has been conjectured that a hierarchy of states appears at low-temperature/poor solvent conditions
where a polymer exists in a finite number of layers close to a surface. We find re-entrant behaviour from the
stretched phase into these layering phases when an appropriate force is applied to the polymer.
Of interest is that the existence, and extent, of this re-entrant phase can be controlled
not only by the force, but also by the ratio of surface-attraction to self-attraction.
We also find that, contrary to what may be expected, the polymer desorbs from the surface
when a sufficiently strong critical force is applied and does not transcend through either
a series of de-layering transitions or monomer-by-monomer transitions.
We discuss the problem mainly from the point of view of the stress ensemble.
However, we make some comparisons with the strain ensemble,
showing the broad agreement between the two ensembles while pointing out subtle differences.
Jarek Krawczyk, Aleksander L. Owczarek, Thomas Prellberg and Andrew Rechnitzer
Layering transitions for adsorbing polymers in poor solvents
Europhys. Lett. 70 726–32(2005)
Link | PDF file |
Show/Hide Abstract
An infinite hierarchy of layering transitions exists for model polymers
in solution under poor solvent or low temperatures and near an attractive surface. A flat histogram stochastic growth
algorithm known as FlatPERM has been used on a self- and surface interacting self-avoiding walk model for lengths up to 256.
The associated phases exist as stable equilibria for large though not infinite length polymers and break the conjectured
Surface Attached Globule phase into
a series of phases where a polymer exists in specified layer close to a surface.
We provide a scaling theory for these phases and the first-order transitions between them.
Richard Brak, Aleksander L. Owczarek, Andrew Rechnitzer and Stuart G. Whittington
A directed walk model of a long chain polymer in a slit with attractive walls
J. Phys. A 38 4309–25(2005)
Link | PDF file |
arXiv |
Show/Hide Abstract
We present the exact solutions of various directed
walk models of polymers confined to a slit and interacting with the walls of the slit via an attractive potential.
We consider three geometric constraints on the ends of the polymer and concentrate on the long chain limit.
Apart from the general interest in the effect of geometrical confinement,
this can be viewed as a two-dimensional model of steric stabilization and
sensitized flocculation of colloidal dispersions. We demonstrate that the
large width limit admits a phase diagram that is markedly different from
the one found in a half-plane geometry, even when the polymer is constrained
to be fixed at both ends on one wall. We are not able to find a closed form
solution for the free energy for finite width, at all values of the interaction
parameters, but we can calculate the asymptotic behaviour for large widths everywhere in the phase plane.
This allows us to find the force between the walls induced by the polymer and hence the regions of the plane where
either steric stabilization or sensitized flocculation would occur.
Buks van Rensburg, Enzo Orlandini, Aleksander L. Owczarek, Andrew Rechnitzer and Stuart G.Whittington
Self-avoiding walks in a slab with attractive walls
J. Phys. A 38 L823–8 (2005)
Link | PDF file |
Show/Hide Abstract
We consider a self-avoiding walk confined between two parallel planes (or lines),
with an energy term associated with each vertex of the walk in the confining planes. We allow the
energy terms to be different for the top and bottom planes. We use exact enumeration and Monte Carlo
methods to investigate the force between the confining planes and how it depends on the width of the slab
and on the interaction energy terms.
The phase diagram is qualitatively similar to that found for a directed walk model.
Jarek Krawczyk, Aleksander L. Owczarek, Thomas Prellberg and Andrew Rechnitzer
Stretching of a chain polymer adsorbed at a surface
J. Stat. Mech.: Theor. Exp. P10004 (2004)
Link | PDF file |
arXiv |
Show/Hide Abstract
In this paper we present simulations of a surface-adsorbed polymer
subject to an elongation force. The polymer is modelled by a self-avoiding walk on a regular lattice.
It is confined to a half-space by an adsorbing surface with attractions for every vertex of the walk visiting the surface,
and the last vertex is pulled perpendicular to the surface by a force. Using the recently proposed flatPERM algorithm,
we calculate the phase diagram for a vast range of temperatures and forces.
The strength of this algorithm is that it computes the complete density of
states from one single simulation. We simulate systems of sizes up to 256 steps.
Henry L. Wong and Aleksander L. Owczarek
Monte Carlo simulations of two-step restricted self-avoiding walks
J. Phys. A. 36 9635–46 (2003)
Link | PDF file |
Show/Hide Abstract
Two-step restricted walk (TSRW) models are a class of restricted self-avoiding walk (SAW) where,
in addition to the self-avoidance constraint, certain restrictions are placed upon each pair of
successive steps. In this paper, we explore the relationship between the restrictions and the
scaling of the average size of walks in three-dimensional models. We use the Pruned-enriched
Rosenbluth method algorithm to perform Monte Carlo studies in five representative TSRW models
in three dimensions. The results present strong numerical evidence that all non-trivial
TSRW models in three dimensions have the same size scaling behaviour as unrestricted SAWs.
This is in contrast to two dimensions where several universality classes are accepted to
exist. In particular, we find no rule analogous to the 'spiral' walk of two dimensions.
Elizabeth A. Beamond, Aleksander L. Owczarek and John Cardy
Quantum and classical localisation and the Manhattan lattice
J. Phys. A. 36 10251–67 (2003)
Link | PDF file |
arXiv |
Show/Hide Abstract
We consider a network model, embedded on the Manhattan lattice,
of a quantum localization problem belonging to symmetry class C. This arises in the context of quasiparticle
dynamics in disordered spin-singlet superconductors which are invariant under spin rotations but not under time reversal.
A mapping exists between problems belonging to this symmetry class and certain classical random walks which are
self-avoiding and have attractive interactions; we exploit this equivalence, using a study of the classical
random walks to gain information about the corresponding quantum problem. In a field-theoretic approach,
we show that the interactions may flow to one of two possible strong-coupling regimes separated by a transition:
however, using Monte Carlo simulations we
show that the walks are in fact always compact two-dimensional objects with a well-defined one-dimensional surface,
indicating that the corresponding quantum system is localized.
Aleksander L. Owczarek and Thomas Prellberg
Monte Carlo Investigation of Lattice Models of Polymer Collapse in Five Dimensions
Int. J. Mod. Phys. C. 14 621–33 (2003)
Link | PDF file |
arXiv |
Show/Hide Abstract
Monte Carlo simulations, using the PERM algorithm,
of interacting self-avoiding walks (ISAW) and interacting self-avoiding trails (ISAT) in
five dimensions are presented which locate the collapse phase transition in those models.
It is argued that the appearance of a transition (at least) as strong as a pseudo-first-order
transition occurs in both models. The values of various theoretically-conjectured dimension-dependent
exponents are shown to be consistent with the data obtained. Indeed the first-order nature of the
transition is even stronger in five dimensions than four. The agreement with the theory is better
for ISAW than ISAT and it cannot be ruled out that ISAT have a true first-order transition in dimension five.
This latter difference would be intriguing if true. On the other hand, since simulations are more difficult
for ISAT than ISAW at this transition in high dimensions, any discrepancy may well be due to the inability
of the simulations to reach the true asymptotic regime.
Aleksander L. Owczarek and Thomas Prellberg
Scaling near the θ-point for isolated polymers in solution
Phys. Rev. E. 67 032801 (2003)
Link | PDF file |
Show/Hide Abstract
Recently questions have been raised as to the
conclusions that can be drawn from currently proposed scaling theory for a single polymer in various
types of solution in two and three dimensions. Here we summarize the crossover theory predicted for
low dimensions and clarify the scaling arguments that relate thermal exponents for quantities
on approaching the 𝜃 point from low temperatures
to those associated with the asymptotics in polymer length at the 𝜃 point itself.
Catherine M. Owczarek, Aleksander L. Owczarek and Phil G. Board
Identification and characterization of polymorphisms at the HSA alpha(1)-acid glycoprotein (ORM) gene locus in Caucasians
Genet. Mol. Biol. 25 13–9 (2002)
Link | PDF file |
Show/Hide Abstract
Human a1-acid glycoprotein (AGP) or orosomucoid (ORM) is a major acute
phase protein that is thought to play a crucial role in maintaining homeostasis. Human AGP is the product of a
cluster of at least two adjacent genes located on HSA chromosome 9.
Using a range of restriction endonucleases we have investigated DNA variation at the locus encoding the AGP genes
in a panel of healthy Caucasians. Polymorphisms were identified using BamHI, EcoRI, BglII, PvuII, HindIII, TaqI and MspI.
Non-random associations were found between the BamHI, EcoRI, BglII RFLPs. The RFLPs detected with PvuII, TaqI and MspI were
all located in exon 6 of both AGP genes. The duplication of an AGP gene was observed in 11% of the indiviuals studied and
was in linkage disequilibrium with the TaqI RFLP.
The identification and characterization of these polymorphisms will prove useful for other population and forensic studies.
Andrew C. Oppenheim, Aleksander L. Owczarek and Richard Brak
Anisotropic step, surface contact, and area weighted directed walks on the triangular lattice
Int. J. Phys. B. 16 1269–99 (2002)
Link | PDF file |
Show/Hide Abstract
We present results for the generating functions of single
fully-directed walks on the triangular lattice, enumerated according to each type of step and weighted
proportional to the area between the walk and the surface of a half-plane (wall), and the number of
contacts made with the wall. We also give explicit formulae for total area generating functions,
that is when the area is summed over all configurations with a given perimeter, and the generating
function of the moments of heights above the wall (the first of which is the total area).
These results generalise and summarise nearly all known results on the square lattice: all
the square lattice results can be obtaining by setting one of the step weights to zero. Our
results also contain as special cases those that already exist for the triangular lattice.
In deriving some of the new results we utilise the Enumerating Combinatorial Objects (ECO)
and marked area methods of combinatorics for obtaining functional equations in the most general cases.
In several cases we give our results both in terms of ratios of infinite q-series and as continued fractions.
Andrew C. Oppenheim, Aleksander L. Owczarek and Richard Brak
Anisotropic step, mutual contact and area weighted festoons and parallelogram polyominoes on the triangular lattice
J. Phys. A 35 3213–30 (2002)
Link | PDF file |
Show/Hide Abstract
We present results for the generating functions of polygons and more
general objects that can touch, constructed from two fully directed walks on the infinite triangular lattice,
enumerated according to each type of step and weighted proportional to the area and the number of contacts
between the directed sides of the objects. In general these directed objects are known as festoons,
being constructed from the so-called friendly directed walks, while the subset constructed from vicious walks are staircase polygons,
also known as parallelogram polyominoes. Additionally, we give explicit formulae for various first area-moment generating functions,
that is when the area is summed over all configurations with a given perimeter. These results generalize and summarize nearly all known results on the square lattice, since such results can be obtained by setting one of the step weights to zero. All our results for the triangular lattice are new and hence provide the opportunity
to study subtle changes in scaling between lattices. In most cases we give our results both in terms of ratios
of infinite q-series and as continued fractions.
Thomas Prellberg and Aleksander L. Owczarek
Pseudo-first-order transition in interacting self-avoiding walks and trails
Comp. Phys. Commun. 147 629–32 (2002)
Link | PDF file |
arXiv |
Show/Hide Abstract
The coil–globule transition of an isolated polymer h
as been well established to be a second-order phase transition described by a standard tri-critical O(0) field theory.
We present Monte Carlo simulations of interacting self-avoiding walks and interacting self-avoiding
trails in four dimensions which provide compelling evidence that the approach to this (tri)critical
point is dominated by the build-up of first-order-like
singularities masking the second-order nature of the coil–globule transition.
Anthony J. Guttmann, Iwan Jensen and Aleksander L. Owczarek
Polygonal polyominoes on the square lattice
J. Phys. A. 34 3721–33 (2001)
Link | PDF file |
Show/Hide Abstract
We study a proper subset of polyominoes, called polygonal polyominoes,
which are defined to be self-avoiding polygons containing any number of holes, each of which is a self-avoiding polygon.
The staircase polygon subset, with staircase holes, is also discussed. The internal holes have no
common vertices with each other, nor any common vertices with the surrounding polygon.
There are no `holes-within-holes'. We use the finite-lattice method to count the number
of polygonal polyominoes on the square lattice. Series have been derived for both the
perimeter and area generating functions. It is known that while the critical point is
unchanged by a finite number of holes, when the number of holes is unrestricted the
critical point changes. The area generating function coefficients grow exponentially,
with a growth constant greater than that for polygons with a finite number of holes,
but less than that of polyominoes. We provide an estimate for this growth constant and
prove that it is strictly less than that for polyominoes. Also, we prove that, enumerating by perimeter,
the generating function of polygonal polyominoes has zero radius of convergence and
furthermore we calculate the dominant asymptotics of its coefficients using rigorous bounds.
Aleksander L. Owczarek, John Essam and Richard Brak
Scaling analysis for the adsorption transition in a watermelon network of n directed non-intersecting walks
J. Stat. Phys. 102 997–1017 (2001)
Link | PDF file |
Show/Hide Abstract
The partition function for the problem of n
directed non-intersecting walks interacting via contact potentials with a wall parallel
to the direction of the walks has previously been calculated as an n×n determinant. Here,
we describe how to analyse the scaling behaviour of this problem using alternative representations
of the solution. In doing so we derive the asymptotics of the partition function of a watermelon
network of n such walks for all temperatures, and so calculate the
associated network exponents in the three regimes: desorbed, adsorbed, and at the adsorption transition.
Furthermore, we derive the full scaling function around the adsorption transition for all n. At the
adsorption transition we also derive a simple “product form” for the partition function. These results have, in part,
been derived using recurrence relations satisfied by the original determinantal solution.
Aleksander L. Owczarek and Thomas Prellberg
Scaling of Self-Avoiding Walks in High Dimensions
J. Phys. A. 34 5773–80 (2001)
Link | PDF file |
arXiv |
Show/Hide Abstract
We examine self-avoiding walks in dimensions 4 to 8 using
high-precision Monte Carlo simulations up to length N = 16 384, providing the first such results in dimensions
d > 4 on which we concentrate our analysis. We analyse the scaling behaviour of the partition function
and the statistics of nearest-neighbour contacts, as well as the average geometric size of the walks, and
compare our results to 1/d-expansions and to excellent rigorous bounds that exist. In particular,
we obtain precise values for the connective constants, µ5 = 8.838 544(3), µ6 = 10.878 094(4), µ7 = 12.902 817(3), µ8 = 14.919 257(2)
and give a revised estimate of µ4 = 6.774 043(5). All of these are by at least one order of magnitude more
accurate than those previously given (from other approaches in d > 4 and all approaches in d = 4).
Our results are consistent with most theoretical predictions, though in d = 5 we find clear evidence of
anomalous N^{-1/2}-corrections for the scaling of the
geometric size of the walks, which we understand as a non-analytic correction to scaling of the general
form N^{(4-d)/2} (not present in pure Gaussian random walks).
Aleksander L. Owczarek, Andrew Rechnitzer and L. Henry Wong
Addendum to `On Three-dimensional Self-avoiding Walk Symmetry Classes’
J. Phys. A. 34 6055–60 (2001)
Link | PDF file |
Show/Hide Abstract
In two dimensions the universality
classes of self-avoiding walks on the square lattice, restricted by allowing only
certain two-step configurations to occur within each walk, has been argued to be
determined primarily by the symmetry of the set of allowed two-step configurations.
In a recent paper (Rechnitzer A and Owczarek A L 2000 On three-dimensional self-avoiding
walk symmetry classes J. Phys. A : Math. Gen. 33 2685-723), primarily tackling the three-dimensional
analogues of these models, a novel two-dimensional model was discovered that seemed either to break
the classification of the models into universality classes according to microscopic symmetry or was
itself a member of a novel universality class. This was supported by series analysis of exact enumeration data.
Here we provide conclusive evidence that this model, known as `anti-spiral walks',
is in the directed walk universality class. We arrive at these conclusions from
Monte Carlo simulations of these walks using a PERM algorithm modified for this problem.
We point out that the behaviour of this model is unusual in that other models in the
directed walk universality class remain directed when the self-avoidance condition is
removed, whereas the behaviour of anti-spiral walks becomes that of a isotropic simple
random walk. We also remark that the symmetry classification of walk models can be kept by
adding a natural condition to the scheme that disallows models, all of whose
configurations avoid some infinite region of the plane by virtue of their microscopic constraints.
Thomas Prellberg and Aleksander L. Owczarek
Four-dimensional polymer collapse II: Interacting self-avoiding trails
Physica A 297 275–90 (2001)
Link | PDF file |
arXiv |
Show/Hide Abstract
We have simulated four-dimensional interacting self-avoiding trails (ISAT)
on the hypercubic lattice with standard interactions at a wide range of temperatures up to length 4096 and at some
temperatures up to length 16384. The results confirm the earlier prediction (using data from a non-standard model
at a single temperature) of a collapse phase transition occurring at finite temperature. Moreover they are in accord
with the phenomenological theory originally proposed by Lifshitz, Grosberg and Khokhlov in three dimensions and
recently given new impetus by its use in the description of simulational results for four-dimensional interacting
self-avoiding walks (ISAW). In fact, we argue that the available data is consistent with the conclusion that the
collapse transitions of ISAT and ISAW lie in the same universality class, in contradiction with long-standing predictions.
We deduce that there exists a pseudo-first order transition for ISAT in four dimensions at finite lengths while the thermodynamic limit is described
by the standard polymer mean-field theory (giving a second-order transition), in contradiction to the prediction
that the upper critical dimension for ISAT is du=4.
Andrew Rechnitzer and Aleksander L. Owczarek
On three-dimensional Self-avoiding walk symmetry classes
J. Phys. A. 33 2685–723 (2000)
Link | PDF file |
Show/Hide Abstract
In two dimensions the universality classes of self-avoiding walks (SAWs)
on the square lattice, restricted by allowing only certain two-step configurations (TSCs) to occur within each walk,
has been argued to be determined primarily by the symmetry of the set of allowed rules.
In three dimensions early work tentatively found one (undirected) universality class different
to that of unrestricted SAWs on the simple cubic lattice. This rule was a natural generalization
of the square lattice `spiral' SAW to three dimensions. In this report we examine a variety of three-dimensional
SAW models with different step restrictions, carefully chosen so as to search for a connection between the symmetry
of the rules and possible new universality classes. A first analysis of the scaling of the
radius of gyration suggests several universality classes, including the one found earlier,
and perhaps some novel class(es). However, a classification of these universality classes
using the symmetries of the rules, or other basic rule properties, is not evident.
Further analysis of the number of configurations
and moment of inertia tensor suggests that in three dimensions the only non-trivial or
undirected universality class is that of unrestricted SAWs.
Thomas Prellberg and Aleksander L. Owczarek
Four-dimensional polymer collapse: Pseudo-First-Order Transition in Interacting Self-avoiding Walks
Phys. Rev. E. 62 3780–9 (2000)
Link | PDF file |
arXiv |
Show/Hide Abstract
In an earlier work we provided the first evidence that the collapse, or coil-globule
transition of an isolated polymer in solution can be seen in a four-dimensional model. Here we investigate, via Monte Carlo simulations,
the canonical lattice model of polymer collapse, namely, interacting self-avoiding walks, to show that it not
only has a distinct collapse transition at finite temperature but that for any finite polymer length this collapse
has many characteristics of a rounded first-order phase transition. However, we also show that there exists a “𝜃 point”
where the polymer behaves in a simple Gaussian manner (which is a critical state), to which these finite-size transition
temperatures approach as the polymer length is increased. The resolution of these seemingly incompatible conclusions
involves the argument that the first-order-like rounded transition is scaled away in the thermodynamic limit to leave
a mean-field second-order transition. Essentially this happens because the finite-size shift of the transition is
asymptotically much larger than the width of the pseudotransition and the latent heat decays to zero (algebraically)
with polymer length. This scenario can be inferred from the application of the theory of Lifshitz, Grosberg, and Khokhlov
(based upon the framework of Lifshitz) to four dimensions: the conclusions of which were written down some time ago by Khokhlov.
In fact it is precisely above the upper critical dimension, which is 3 for this problem, that the theory of Lifshitz may be quantitatively applicable to polymer collapse.
Aleksander L. Owczarek and Thomas Prellberg
First-order scaling approach to a second-order phase transition: Tricritical polymer collapse
Europhys. Lett. 51 602–7 (2000)
Link | PDF file |
arXiv |
Show/Hide Abstract
The coil-globule transition of an isolated polymer has been well
established to be a second-order phase transition described by a standard tricritical O(0) field theory.
We provide compelling evidence from Monte Carlo simulations in four dimensions, where mean-field theory should apply,
that the approach to this (tri)critical point is dominated by the build-up of first-order–like singularities
masking the second-order nature of the coil-globule transition: the distribution of the internal energy having
two clear peaks that become more distinct and sharp as the tricritical point is approached. However,
the distance between the peaks slowly decays to zero. The evidence shows that the position of this (pseudo)
first-order transition is shifted by an amount from the tricritical point that is asymptotically much larger
than the width of the transition region.
Thomas Prellberg and Aleksander L. Owczarek
On the Asymptotics of the Finite-Perimeter Partition Function of Two-Dimensional Lattice Vesicles
Commun. Math. Phys. 201 493–505 (1999)
Link | PDF file |
arXiv |
Show/Hide Abstract
We derive the dominant asymptotic form and the order of the
correction terms of the finite-perimeter partition function of self-avoiding polygons on the square lattice,
which are weighted according to their area A as q A, in
the inflated regime, q >1. The approach q→ 1+ of the asymptotic form is examined.
Richard Brak, John Essam and Aleksander L. Owczarek
Exact solution of N directed non-intersecting walks interacting with one or two boundaries
J. Phys. A. 32 2921–9 (1999)
Link | PDF file |
Show/Hide Abstract
The partition function for the problem of an arbitrary
number of directed non-intersecting walks interacting with one or two walls parallel to the direction
of the walks is calculated exactly utilizing a theorem recently proved concerning the Bethe ansatz for
the eigenvectors of the transfer matrix of the five-vertex model. This theorem shows that the
completeness of the Bethe ansatz eigenvectors for the
N-walk transfer matrix can be deduced from the completeness of the one-walk eigenvectors.
Richard Brak and Aleksander L. Owczarek
Lattice path combinatorial interpretation of the six-vertex free fermion condition
J. Phys. A. 32 3497–504 (1999)
Link | PDF file |
arXiv |
Show/Hide Abstract
The free-fermion condition of the six-vertex model provides a five-parameter sub-manifold on which the
Bethe ansatz equations for the wavenumbers that enter into the eigenfunctions of the transfer matrices of the model decouple,
hence allowing explicit solutions. Such conditions arose originally in early field-theoretic S-matrix approaches.
Here we provide a combinatorial explanation for the condition in terms of a generalized Gessel-Viennot involution. By doing so we extend the use of the Gessel-Viennot theorem, originally devised for non-intersecting walks only, to a special
weighted type of intersecting walk, and hence express the partition function of N such walks starting and finishing at fixed endpoints in terms of the single-walk partition functions.
Richard Brak, John Essam and Aleksander L. Owczarek
From the Bethe Ansatz to the Gessel-Viennot Theorem
Annals of Comb. 3 251–63 (1999)
Link | PDF file |
arXiv |
Show/Hide Abstract
We state and prove several theorems that
demonstrate how the coordinate Bethe Ansatz for the eigenvectors of suitable transfer matrices
of a generalized inhomogeneous, five-vertex model on the square lattice,
given certain conditions hold, is equivalent to the Gessel-Viennot determinant for the
number of configurations ofN non-intersecting directed lattice paths, or vicious walkers,
with various boundary conditions.
Our theorems are sufficiently general to allow generalisation to any regular planar lattice.
Richard Brak, John Essam and Aleksander L. Owczarek
Partial Difference equation method for lattice path problems
Annals of Comb. 3 265–75 (1999)
Link | PDF file |
Show/Hide Abstract
Many problems concerning lattice paths, especially on the
square lattice have been exactly solved. For a single path, many methods exist that allow exact calculation
regardless of whether the path inhabits a strip, a semi-infinite space or infinite space, or perhaps interacts with the walls.
It has been shown that a transfer matrix method using the Bethe Ansatz allows for the
calculation of the partition function for many non-intersecting paths interacting with a wall.
This problem can also be considered using the Gessel-Viennot methodology.
In a concurrent development, two non-intersecting paths interacting with a wall
have been examined in semi-infinite space using a set of partial difference equations.
Here, we review thispartial difference equation method for the case of one path in a half plane.
We then demonstrate that the answer for arbitrary numbers of non-intersecting paths interacting
with a wall can be obtained using this method. One reason for doing this is its pedagogical
value in showing its ease of use compared to the transfer matrix method.
The solution is expressed in a new form as a “constant term” formula,
which is readily evaluated. More importantly, it is the natural method that generalizes easily to many
intersecting paths where there is inter-path interactions (e.g., osculating lattice paths).
We discuss the relationship of the partial difference equation method to the transfer matrix
method and their solution via a Bethe Ansatz.
Deborah Bennett-Wood, Ian G. Enting, David S. Gaunt, Anthony J. Guttmann, Jane L. Leask, Aleksander L. Owczarek and Stu G. Whittington
Exact Enumeration Study of Free Energies of Interacting Polygons and Walks in Two Dimensions
J. Phys. A. 31 4725–41 (1998)
Link | PDF file |
arXiv |
Show/Hide Abstract
We present analyses of substantially extended series for both
interacting self-avoiding walks (ISAW) and polygons (ISAP) on the square lattice. We argue that these provide
good evidence that the free energies of both linear and ring polymers are equal above the -temperature, thus
extending the application of a theorem of Tesi et al to two dimensions. Below the -temperature the conditions of
this theorem break down, in contradistinction to three dimensions, but an analysis of the ratio of the partition
functions for ISAP and ISAW indicates that the free energies are in fact equal at all temperatures within at least.
Any perceived difference can be interpreted as the difference in the size of corrections to scaling in both problems.
This may be used to explain the vastly different values of the crossover exponent previously estimated for ISAP to
that predicted theoretically, and numerically confirmed, for ISAW. An analysis of newly extended neighbour-avoiding
self-avoiding walk series is also given.
Richard Brak, Aleksander L. Owczarek and Chris Soteros
On Anisotropic Spiral Self-avoiding Walks
J. Phys. A. 31 4851–69 (1998)
Link | PDF file |
Show/Hide Abstract
We report on a Monte Carlo study of so-called two-choice-spiral
self-avoiding walks on the square lattice. These have the property that their geometric size (such as is
measured by the radius of gyration) scales anisotropically, with exponent values that seem to defy rational
fraction conjectures. This polymer model was previously understood to be in a universality class different
to ordinary self-avoiding walks, directed walks (which are also anisotropic), and symmetric spiral walks,
in two dimensions. Our Monte Carlo study concurs with those previous exact enumeration studies in that respect.
However, we estimate substantially different values for the scaling exponents associated with the
geometric size of the walks. We give arguments that explain this difference in terms of a turning
point in the local exponent values, and in turn explain this by
arguing for the existence of probable logarithmic corrections. We also supply numerical evidence
supporting a conjecture concerning the angle of anisotropy in the model.
Murray T. Batchelor, Deborah Bennett-Wood and Aleksander L. Owczarek
Two-dimensional polymer networks at a mixed boundary: Surface and wedge exponents
Europ. Phys. J. B 5 139–42 (1998)
Link | PDF file |
arXiv |
Show/Hide Abstract
We present analyses of substantially extended series for
both interacting self-avoiding walks (ISAW) and polygons (ISAP) on the square lattice.
We argue that these provide good evidence that the free energies of both linear and ring polymers are equal above the Θ-temperature,
thus extending the application of a theorem of Tesi et al to two dimensions. Below the Θ-temperature the conditions of this theorem break down, in contradistinction to three dimensions, but an analysis of the ratio of the partition functions for ISAP and ISAW indicates that the free energies are in fact equal at all temperatures within at least. Any perceived difference can be interpreted as the difference in the size of corrections to scaling in both problems. This may be used to explain the vastly different values of the crossover exponent previously estimated for ISAP to that predicted theoretically, and numerically confirmed, for ISAW.
An analysis of newly extended neighbour-avoiding self-avoiding walk series is also given.
Deborah Bennett-Wood, John L. Cardy, Ian G. Enting, Anthony J. Guttmann and Aleksander L. Owczarek
On the Non-Universality of a Critical Exponent for Self-Avoiding Walks
Nuc. Phys. B 528 533–52 (1998)
Link | PDF file |
arXiv |
Show/Hide Abstract
We have extended the enumeration of self-avoiding walks on the
Manhattan lattice from 28 to 53 steps and for self-avoiding polygons from 48 to 84 steps.
Analysis of this data suggests that the walk generating function exponent γ = 1.3385 ± 0.003,
which is different from the corresponding exponent on the square, triangular and honeycomb lattices.
This provides numerical support for an argument recently advanced by Cardy,
to the effect that excluding walks with parallel nearest-neighbour steps should cause a change in the exponent γ.
The lattice topology of the Manhattan lattice precludes such parallel steps.
Richard Brak, Paul P. Nidras and Aleksander L. Owczarek
Cluster Structure of Collapsing Polymers
J. Stat. Phys. 91 75–93 (1998)
Link | PDF file |
Show/Hide Abstract
In order to better understand the geometry of the polymer collapse transition,
we study the distribution of geometric clusters made up of the nearest neighbor interactions of an interacting self-avoiding walk.
We argue for this new correlated percolation problem that in two dimensions, and possibly
also in three dimensions, a percolation transition takes place at a temperature lower than the
collapse transition. Hence this novel transition should be governed by exponents unrelated to the θ-point exponents.
This also implies that there is a temperature range in which the polymer has collapsed, but has no long-range cluster
structure. We use Monte Carlo to study the distribution of clusters on the simple cubic and Manhattan lattices.
On the Manhattan lattice, where the data are most convincing, we find that the percolation transition occurs
at ω p =1.461(3), while the collapse transition is known to occur exactly at ω θ =1.414....
We propose a finite-size scaling form for the
cluster distribution and estimate several of the critical exponents. Regardless of the value of ω p ,
this percolation problem sheds new light on polymer collapse.
Aleksander L. Owczarek and Thomas Prellberg
Existence of four-dimensional polymer collapse I: Kinetic growth trails
Physica A 260 20–30 (1998)
Link | PDF file |
Show/Hide Abstract
We present the results of simulations of kinetic growth trails (KGT)
(bond-avoiding walks) in four dimensions. We use a mapping from a kinetic growth model to a static model of
self-interacting trails (ISAT) at a particular temperature to argue that this temperature is precisely the
collapse temperature of four-dimensional interacting trails. To do this we show that the kinetic growth trails
behave neither like static non-interacting trails, which should behave as excluded-volume-dominated
four-dimensional polymers (that is self-avoiding walks), or collapsed four-dimensional polymers,
but rather show an intermediate behaviour. This is the first indication of collapse in any
four-dimensional lattice polymer model and so may be helpful in deciding which of the competing models
of polymers is a good model in lower dimensions. We have calculated various exponents of the KGT model a
nd identified them with certain critical exponents of the static ISAT problem.
Anthony J. Guttmann, Aleksander L. Owczarek and Xavier G. Viennot
Vicious walkers and Young tableaux I: Without walls
J. Phys. A. 31 8123–35 (1998)
Link | PDF file |
Show/Hide Abstract
We rederive previously known results for the number of
star and watermelon configurations by showing that these follow immediately from standard results in
the theory of Young tableaux and integer partitions. In this way
we provide a proof of a result, previously only conjectured, for the total number of stars.
Richard Brak, John Essam and Aleksander L. Owczarek
New Results for Directed Vesicles and Chains near an Attractive Wall
J. Stat. Phys. 93 155–92 (1998)
Link | PDF file |
Show/Hide Abstract
In this paper we present new exact results for single fully directed walks and
fully directed vesicles near an attractive wall. This involves a novel method of solution for these types of problems.
The major advantage of this method is that it, unlike many other single-walker methods, generalizes to an arbitrary number of walkers.
The method of solution involves solving a set of partial difference equations with a Bethe Ansatz.
The solution is expressed as a “constant-term” formula which evaluates to sums of products of binomial coefficients.
The vesicle critical temperature is found at which a binding transition takes place, and the asymptotic forms of the a
ssociated partition functions are found to have three different entropic exponents depending on whether the temperature is above,
below, or at its critical value. The expected number of monomers adsorbed onto the surface is found to become proportional to the
vesicle length at temperatures below critical. Scaling functions near the critical point are determined.
Aleksander L. Owczarek, Andrew Rechnitzer, Richard Brak and Anthony J. Guttmann
On the Hulls of Directed Percolation Clusters
J. Phys. A. 30 6679–91 (1997)
Link | PDF file |
Show/Hide Abstract
The properties of the hulls of directed percolation clusters are studied.
The scaling and finite-size scaling of many quantities around the percolation threshold are derived and a novel Monte Carlo algorithm,
which is more than twice as fast as the standard algorithm at p_c, has been formulated to study these properties.
Simulations have been conducted that enable an estimation of all the exponents involved. In particular, the central exponent, x, relating the average hull length of clusters to their mass,
has been estimated to be 0.773(4) at the percolation threshold. This same exponent is estimated to be 0.905(5) for p< p_c>.
Thus, this second value should hold for directed animals.
Deborah Bennett-Wood and Aleksander L. Owczarek
Exact Enumeration Results for Self-Avoiding Walks on the Honeycomb Lattice Attached to a Surface
J. Phys. A. 29 4755–68 (1996)
Link | PDF file |
Show/Hide Abstract
We consider self-avoiding walks on the honeycomb lattice interacting
with a surface with different energies associated between sites in contact with a linear boundary
to the left of the origin and those in contact with the right of the boundary.
We numerically confirm recent exact results for the polymer adsorption transition
and corresponding critical exponents with mixed ordinary and special boundary conditions.
The phase diagram is elucidated with the aid of some rigorous arguments.
Murray T. Batchelor, Aleksander L. Owczarek, Katherine Seaton and C. M. Yung
Surface Critical Behaviour of an O(n) Loop Model Related to Two Manhattan Lattice Walk Problems
J. Phys. A 28 839–52 (1995)
Link | PDF file |
Show/Hide Abstract
We find and discuss the scaling dimensions of the branch
0 manifold of the Nienhuis O(n) loop model on the square lattice, concentrating on the surface dimensions.
The results are extracted from a Bethe ansatz calculation of the finite-size corrections to the eigenspectrum
of the six-vertex model with free boundary conditions. These results are especially interesting for
polymer physics at two values of the crossing parameter lambda . Interacting self-avoiding walks on the
Manhattan lattice at the collapse temperature ( lambda = pi /3) and Hamiltonian walks on the Manhattan lattice
( lambda = pi /2) are discussed in detail.
Our calculations illustrate the importance of examining both odd and even strip widths when performing
finite-size correction calculations to obtain scaling dimensions.
Thomas Prellberg and Aleksander L. Owczarek
Models of Polymer Collapse in Three Dimensions: Evidence from Kinetic Growth Simulations.
Phys. Rev. E. 51 2142–9 (1995)
Link | PDF file |
Show/Hide Abstract
We present simulational evidence that kinetically
grown tricolor walks and kinetically grown trails on selected lattices in three dimensions are
equivalent to interacting walks and trails at their respective collapse temperatures,
all of which model polymers in dilute solution at the θ point. We discuss the relation of these
models to the canonical model of a single self-attracting polymer in dilute solution: the self-avoiding walk
with nearest-neighbor interactions. The main results concern the divergence of the specific heat
and these differ from the predictions of the (three-parameter) Edwards model.
The behavior of the kinetic trail simulations also raises doubts about the
current field theoretic description of collapse in interacting trails.
Aleksander L. Owczarek and Thomas Prellberg
The Collapse Point of Interacting Trails in Two Dimensions from Kinetic Growth Simulations
J. Stat. Phys. 79 951–67 (1995)
Link | PDF file |
Show/Hide Abstract
We present simulational evidence that kinetic growth trails
on the square lattice are equivalent to interacting trails at their collapse temperature.
As a consequence we give values for most of the canonical exponents of the trail collapse transition:
these are significantly different from those proposed for interacting walks. We can also
interpret our results in terms of the equivalent Lorentz lattice gas
and find that this model does not display diffusion, as has been previously thought. Rather,
the mean square displacement grows as t logt in time t.
Thomas Prellberg and Aleksander L. Owczarek
Stacking Models of Vesicles and Compact Clusters
J. Stat. Phys. 80 755–79 (1995)
Link | PDF file |
Show/Hide Abstract
We investigate three simple lattice models of two dimensional vesicles.
These models differ in their behavior from the universality class of partially convex polygons, which has been recently established.
They do not have the tricritical scaling of those models, and furthermore display a surprising feature: their (perimeter)
free energy is discontinuous with an isolated value at zero pressure. We give the full
asymptotic descriptions of the generating functions in area and perimeter variables from the q-series solutions and obtain the scaling functions where applicable.
Richard Brak and Aleksander L. Owczarek
On the Analyticity Properties of Scaling Functions in Models of Polymer Collapse
J. Phys. A. 28 4709–25 (1995)
Link | PDF file |
Show/Hide Abstract
We consider the mathematical properties of the generating and
partition functions in the two-variable scaling region about the tricritical point in some models of
polymer collapse. We concentrate on models that have a similar behaviour to that of interacting
partially-directed self-avoiding walks (IPDSAW) in two dimensions. However, we do not restrict the
discussion to that model. After describing the properties for a general class of models, and stating
exactly what we mean by scaling, we prove the following theorem: If the generating function of
finite-size partition functions has a tricritical cross-over scaling form around the theta -point,
and the associated tricritical scaling function, g, has a finite radius of convergence, then the
partition function has a finite-size scaling form and importantly the finite-size scaling function, f,
is an entire function. In the IPDSAW case we have an explicit representation of the finite-size scaling function.
We point out that given our description of tricritical scaling this theorem should apply
mutatis mutandis to a wider class of Θ-point models.
We discuss the result in relation to the Edwards model of polymer collapse for which it has recently been argued that the finite-size scaling functions are not entire.
Deborah Bennett-Wood, John L. Cardy, Silvia Flesia, Anthony J. Guttmann and Aleksander L. Owczarek
Oriented Self-Avoiding Walks with Orientation-Dependent Interactions
J. Phys. A. 28 5143–63 (1995)
Link | PDF file |
Show/Hide Abstract
We consider oriented self-avoiding walks on the square lattice
with different energies between steps that are oriented parallel or antiparallel across a face of the lattice.
Rigorous bounds on the free energy and exact enumeration data are used to study the statistical mechanics of this model.
We conjecture a phase diagram in the parallel-antiparallel interaction plane, and discuss the order of the associated
phase transitions. The question, raised by previous field theoretical considerations, of the existence of an exponent
that varies continuously with the energy of interaction is discussed at length. In connection with this we have also
studied two oriented walks fixed at a common origin;
this being the simplest model of branched oriented polymers in two dimensions. The evidence, although not conclusive, tends to support the field theoretic prediction.
Deborah Bennett-Wood, Richard Brak, Anthony J. Guttmann, Aleksander L. Owczarek and Thomas Prellberg
Low Temperature Partition Function Scaling: Series Analysis Results
J. Phys. A. 27 L1–8 (1994)
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By utilizing newly extended series for self-avoiding walks and polygons
with nearest-neighbour interactions on the square lattice we have examined the validity of a recent conjecture
on the scaling of their partition functions at low temperatures. The ratio of the walk to polygon partition functions
should have a length-dependent power law singularity, ngamma (D) at all temperatures. At low temperatures we find
gamma D is 0.92+or-0.09 in distinction to the conjectured value of 19/16=1.1875, though we find agreement at
high temperatures and at the theta -temperatures with the conjectured values there.
Thomas Prellberg and Aleksander L. Owczarek
Manhattan Lattice Θ-point Exponents from Kinetic Growth Walks and Exact Results from the Nienhuis O(n) Model
J. Phys. A. 27 1811–26 (1994)
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Kinetic growth walks (KGW) on the Manhattan lattice
have previously been shown to be equivalent to the static problem of interacting self-avoiding walks
on that lattice at the Θ-temperature. Here, we illustrate how a complete set of exponents for the
static problem, including the crossover exponent phi and surface exponents at the ordinary and special
transition points, may be obtained from simulations of kinetic walks. In the process we find that
φ ≃ 0.430+or-0.006 which encompasses the conjectured value 3/7 ≃ 0.42857 for the theta -point
on an isotropic lattice. Our numerics confirm a predicted set of exponents for both the bulk and surface
transitions in addition to results such as the exact internal energy at the bulk transition. Furthermore,
we point out that a recently examined variant of the Nienhuis O(n) model can be mapped onto
Θ-point walks on the Manhattan lattice which allows identification of the scaling dimensions for that problem and thereby provides a method for proving all the numerical conjectures.
Deborah Bennett-Wood, Aleksander L. Owczarek and Thomas Prellberg
Crossover in Smart Kinetic Growth Walks
Physica A. 206 283–9 (1994)
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By extending the study of smart kinetic growth walks on the honeycomb lattice
we are able to extract a value for the crossover exponent, 0.44 ± 0.02,
which is consistent with the conjectured value of 3/7 ≈ 0.428 for interacting walks at the θ-temperature.
This work corroborates similar results on the oriented Manhattan lattice.
Aleksander L. Owczarek and Thomas Prellberg
Interacting Partially Directed Walks: A Model of Polymer Collapse.
Physica A 205 203–13 (1994)
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A survey of the results that have been obtained to date on the
partially directed walk model of the polymer collapse transition is presented.
Richard Brak, Aleksander L. Owczarek and Thomas Prellberg
Exact Scaling Behaviour of Partially Convex Vesicles
J. Stat. Phys. 76 1101–28 (1994)
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We solve analytically for the perimeter-area generating functions for two models of vesicles.
While from the solution of the first model, staircase polygons, one can easily extract the asymptotic scaling behavior,
the exact solution of the second, column-convex polygons, is difficult to analyze.
This leads us to apply a recently developed method for deriving the scaling behavior
indirectly, utilizing a set of nonlinear differential equations.
One result of this work is a nontrivial confirmation of the scaling/universality hypothesis.
Aleksander L. Owczarek, Thomas Prellberg, Deborah Bennett-Wood and Anthony J. Guttmann
Universal Distance Ratios for Interacting Two-dimensional Polymers
J. Phys. A 27 L919–24 (1994)
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We present the hypothesis that at the theta -temperature of a single
interacting polymer in two dimensions, a particular combination of universal distance amplitude ratios is given
exactly by a simple formula in terms of critical indices only. This generalizes a similar claim for noninteracting
self-avoiding walks, which was derived from conformal invariance considerations. We support our hypothesis with a
series analysis of interacting self-avoiding walks on the square lattice,
high-precision simulations of interacting self-avoiding walks on the Manhattan lattice at the exact theta -point
and similar simulations of interacting trails on the square lattice.
Phillipe M. Binder, David Y. K. Ko, Aleksander L. Owczarek and Carol J. Twining
Ordered Cellular Automata In One-Dimension J. Physique. I 23 21–8 (1993)
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We study a probabilistic one-dimensional majority-rule two-state
cellular automaton and examine the stability of ordered magnetised states in systems of size L as the neighbourhood
radius R varies. We find that a scaling $R \sim \ln L$ is sufficient for an ordered phase to be metastable, i.e.,
to survive for times much longer than the typical critical fluctuation.
The lattice magnetisation obeys a scaling relation which agrees with results from mean-field analysis.
Aleksander L. Owczarek and Thomas Prellberg
Exact Solution of the Discrete (1+1)-Dimensional SOS Model with Field and Surface Interactions
J. Stat. Phys. 70 1175–94 (1993)
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We present the solution of a linear solid-on-solid (SOS) model.
Configurations are partially directed walks on a two-dimensional square lattice and we include a linear surface tension,
a magnetic field, and surface interaction terms in the Hamiltonian. There is a wetting transition at zero field and,
as expected, the behavior is similar to a continuous model solved previously.
The solution is in terms ofq-series most closely related to theq-hypergeometric functions1 φ 1.
Aleksander L. Owczarek, Thomas Prellberg and Richard Brak
A New Scaling Form for the Collapsed Polymer Phase
Phys. Rev. Lett. 70 951–3 (1993)
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By studying the finite length scaling of a self-interacting
partially directed self-avoiding walk we have verified a new scaling form for the collapsed phase of self-avoiding-walk problems.
We suggest therefore that this should hold in polymer systems.
Aleksander L. Owczarek, Thomas Prellberg and Richard Brak
Reply to Exact Scaling Form for the 2D Polymer Phase
Phys. Rev. Lett. 71 4275 (1993)
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Aleksander L. Owczarek
Scaling in the Collapsed Polymer Phase: Exact Results
J. Phys. A. 26 L647–53 (1993)
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The author derives an exact scaling form for the low temperature
partition function in a model of polymer collapse. This confirms series work and so gives the exponents sigma =1/2,
gamma -=1/4 and chi =3/4 exactly. The model considered
is a variant of the self-interacting partially directed self-avoiding walk in two dimensions.
Thomas Prellberg, Aleksander L. Owczarek, Richard Brak and Anthony J. Guttmann
Finite Length Scaling of Collapsing Directed Walks
Phys. Rev. E. 48 2386–96 (1993)
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We study the finite-length scaling of self-interacting
partially directed self-avoiding walks utilizing enumeration data up to a total length of 6000 steps.
This facilitates the evaluation of the numerical techniques available for calculating exponents at the θ (critical) point
and in the collapsed phase of walk-type models. Another consequence is the conjecture of an alternative scaling theory for
the collapsed region of the phase diagram and the suggestion that this should be applicable to the wider range of
undirected problems including interacting self-avoiding walks. We provide a phenomenological picture of the
phase transition in terms of the condensation
of droplets that allows us to understand the various length scales involved in the problem.
Aleksander L. Owczarek, Thomas Prellberg and Richard Brak
The Tricritical Behaviour of Self-Interacting Partially Directed Walks
J. Stat. Phys. 72 737–72 (1993)
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We present the thermodynamics of two variations of the interacting partially
directed self-avoiding walk problem by discussing versions where the length of the walks assume real as well as a integral values.
While the discrete model has been considered previously to varying degrees of success, the continuous model we now define has not.
The examination of the continuous model leads to theexact derivation of several exponents. For the discrete model some of these
exponents can be calculated using a continued-fraction representation. For both models the crossover exponentφ is found to be 2/3.
Moreover, we confirm the tricritical nature of the collapse transition in the generalized ensemble and calculate the full scaling
form of the generating function.
Additionally, the similarities noticed previously to other models, but left unexplored, are explained with the aid of necklacing arguments.
Richard Brak, Aleksander L. Owczarek and Thomas Prellberg
A scaling theory of the collapse transition in geometric cluster models of polymers and vesicles
J. Phys. A. 26 4565–79 (1993)
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Much effort has been expended in the past decade to calculate numerically
the exponents at the collapse transition point in walk, polygon and animal models. The crossover exponent phi has been
of special interest and sometimes is assumed to obey the relation 2- alpha =1/ phi with the alpha the canonical (thermodynamic)
exponent that characterizes the divergence of the specific heat. The reasons for the validity of this relation are not widely known.
The authors present a scaling theory of collapse transitions in such models. The free energy and canonical
partition functions have finite-length scaling forms whilst the grand partition function has a tricritical scaling form.
The link between the grand and canonical ensembles leads to the above scaling relation. They then comment on the validity of
current estimates of the crossover exponent for interacting self-avoiding walks in two dimensions and propose a test involving
the scaling relation which may be used to check these values.
Anthony J. Guttmann, Thomas Prellberg and Aleksander L. Owczarek
On the Symmetry Classes of Planar Self-Avoiding Walks
J. Phys. A. 26 6615–23 (1993)
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We present new results on the class of anisotropic, spiral walks in two dimensions.
We find that these are directed problems in the sense that the usual relation, v||=2vperpendicular to , holds between the
length scale exponents. In contradistinction, however, they do not seem to fall in the usual directed universality class (v||=1).
Motivated by this, the universality classes of self-avoiding walks (SAW) on the square lattice are discussed.
We argue that of the 84 models that exist by
restricting the possible two step configurations there are four major categories with a total of seven generic types.
The importance of reflection symmetry in this classification is discussed.
Aleksander L. Owczarek, Phillipe M. Binder and David Y. K. Ko
Cutoff Scalings for One Dimensional Order
J. Phys. A 25 L21–4 (1992)
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The authors consider one-dimensional Ising systems of length L
with an interaction cut-off R(L) in the limit R to infinity . They present free energy arguments for order a
t low temperatures provided R(L))(A In L)1 kappa /, where kappa is a new exponent;
this opens the possibility of one-dimensional pseudo-order in macroscopic systems.
Joesph O. Indekeu, Aleksander L. Owczarek and Michael R. Swift
Quasiwetting and Critical Point Leaps
Phys. Rev. Letts. 66 2174 (1991)
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Michael R. Swift, Aleksander L. Owczarek and Joesph O. Indekeu
Effect of Confinement on Wetting and Drying between Opposing Boundaries
Europhys. Letts. 14 475–81(1991)
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We study quasi-wetting transitions in confined systems in which
capillary condensation is suppressed. In particular, we are concerned with adsorbates between opposing walls
(one wall favours wetting, the other drying).
We employ an Ising model and calculate the global phase diagram for a slab of width L and boundaries with opposite surface fields,
in Landau theory. We find novel first-order, critical, and tricritical quasi-wetting transitions, which converge smoothly,
for L → ∞, to the familiar wetting transitions. We question the recently proposed novel mechanism for critical-point shifts in films.
Douglas B. Abraham and Aleksander L. Owczarek
Correlation Function for the Baxter Model
Phys. Rev. Lett. 64 2595–8 (1990)
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We derive an exact expression for the low-temperature correlation function
in the eight-vertex-equivalent Ising model. We provide a phenomenological interpretation of the qualitative nature
of the transfer-matrix spectrum in terms
of bubble excitations and speculate about the behavior in nonzero magnetic field.
Phillipe M. Binder, Aleksander L. Owczarek, Andrew R. Veal and Julia M. Yeomans
Collapse Transition in a Simple Polymer Model: Exact Results
J. Phys. A 23 L975–9 (1990)
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Aleksander L. Owczarek and Rodney J. Baxter
Surface Free Energy of the Critical Six Vertex Model with Free Boundaries
J. Phys. A 22 1141–65 (1989)
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The Bethe ansatz equations are derived for the six-vertex model with general boundary
weights on a lattice in a diagonal orientation. These are solved in the thermodynamic limit.
Finite-size corrections to the free energy (for a restricted class of boundary weights,
including those corresponding to a Potts model with free boundaries) are calculated in the critical region.
The first-order term gives the surface free energy of the model. The second-order term is
found to be - pi okT tan( pi nu /2 mu )c/48N'2 where c=(1-6 mu 2/( pi 2- pi mu )) is the conformal anomaly.
This can be compared to - pi kT sin( pi nu / mu )c/6N'2 for a calculation on the lattice in the
standard orientation and with periodic boundary conditions.
This difference can be explained geometrically using conformal invariance.
Aleksander L. Owczarek and Rodney J. Baxter
Generalised Percolation Probabilities for the Self-Dual Potts Model
J. Phys. A 20 5263–71 (1987)
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A set of generalised percolation probabilities
Pn are defined for the dichromatic polynomial formulation of the Potts model.
A generating function for these Pn is calculated at the self-dual temperature.
P1 and P2 are explicitly given and the behaviour of Pn is investigated.
Aleksander L. Owczarek and Rodney J. Baxter
A Class of Interaction-Round-a-Face Models and Its Equivalence with an Ice-Type Model
J. Stat. Phys. 49 1093–115 (1987)
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A new model (called the Temperley-Lieb interactions model) is introduced,
in two-dimensional lattice statistics, on a square lattice ℒ.
The Temperley-Lieb equivalence of this model to the six-vertex, self-dual Potts, critical hard-hexagons and
critical nonintersecting string models is established. A
graphical equivalence of this model to the six-vertex model generalizes
this equivalence to noncritical cases of the above models.
The order parameters of a specialization of this model are studied.
Conference Proceedings (unrefereed)
Chris Bradly and Aleksander L. Owczarek
Adsorption transition of self-avoiding walks on high-dimensional lattices
2023 MATRIX Annals. (Springer) (2025)
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It is known that the critical properties of the adsorption transition for
the self-avoiding walk model of lattice polymers are not superuniversal. In particular
this is because the value of the crossover exponent φ in the three-dimensional
case deviates from the mean-field prediction φ = 1/2. The mean-field value is exactly
true for high-dimensional lattices with dimension d > 4, where d = 4 is the
upper critical dimension. These high-dimensional cases have not been studied extensively
and here we employ Monte Carlo simulations to investigate the scaling of
the adsorbed fraction, finding that the expected mean-field scaling holds for d = 5, 6.
However, we do not find additional logarithmic scaling for d = 4, contrary to what
is typically expected for lattice models at the upper critical dimension.
Anthony J. Guttmann, Aleksander L. Owczarek, Deborah Bennett-Wood and Thomas Prellberg
Recent Developments in the Study of Walks, Polygons and the Ising Model
Nuc. Phys. B. (Proc. Suppl.) 42 911–3 (1995)
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In this paper we discuss some general developments in the
study of self-avoiding walks and polygons in two dimensions, as well Ising models in both two- and three-dimensions,
and then focus in greater detail on universal distance ratios for interacting self-avoiding walks.
Editorships
Richard Brak, Omar Foda, Catherine Greenhill, Anthony J. Guttmann and Aleksander L. Owczarek
Proceedings of the 2002 Formal Power Series and Algebraic Combinatorics Conference (2002)
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