Papers
Most of my papers can be found on my arXiv page. See also my Google Scholar page and my ORCID page.
Preprints
Chris J. Bradly, Nicholas R. Beaton and Aleksander L. Owczarek
Lattice polymers near a permeable interface
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, Mark Holmes and Xin Huang
Chemical distance for the half-orthant model
arXiv | Show/Hide Abstract
The half-orthant model is a partially oriented model of a random medium involving a parameter $p\in [0,1]$, for which there is a critical value $p_c(d)$ (depending on the dimension $d$) below which every point is reachable from the origin. We prove a limit theorem for the graph-distance (or "chemical distance") for this model when $p\lt p_c(2)$ and also when $1-p$ is larger than the critical parameter for site percolation in $\mathbb{Z}^d$. The proof involves an application of the subadditive ergodic theorem. Novel arguments herein include the method of proving that the expected number of steps to reach any given point is finite, as well as an argument that is used to show that the shape is "non-trivial" in certain directions.
Nicholas R. Beaton, Kai Ishihara, Mahshid Atapour, Jeremy W. Eng, Mariel Vazquez, Koya Shimokawa, and Christine E. Soteros
Entanglement statistics of polymers in a lattice tube and unknotting of 4-plats
arXiv | Show/Hide Abstract
From polymer models, it has been conjectured that the exponential growth rate of the number of lattice polygons with knot-type $K$ is the same as that for unknot polygons, and that the entropic critical exponent increases by one for each prime knot factor in the knot decomposition of $K$. Here we prove this conjecture for any knot or non-split link embeddable in $\mathbb{T}^*$, the $\infty\times2\times1$ tubular sublattice of the simple cubic lattice. This is the first model for which the conjecture has been proved. For the proof, we establish upper and lower bounds relating the asymptotics of the number of $n$-edge polygons with fixed knot or link-type $K$ in $\mathbb{T}^*$ (as $n\to\infty$) to that of the number of $n$-edge unknot polygons. For the upper bounds, we prove that polygons of any knot or non-split link type $K$ can be unknotted by $f_K$ braid insertions, where $f_K$ is the number of prime factors of $K$. As part of the proof, we show that a 4-plat diagram of a 2-bridge link can always be unknotted by the insertion of a 4-braid diagram whose crossing number is bounded by the minimal crossing number of the link. For the lower bounds, we prove a pattern theorem for unknots using information from exact transfer-matrix calculations for polygons in the tube. Thus, unknot polygons can be transformed into any fixed knot or non-split link type by inserting one pattern for each prime factor.
Publications
Nicholas R. Beaton, Kai Ishihara, Mahshid Atapour, Jeremy W. Eng, Mariel Vazquez, Koya Shimokawa, and Christine E. Soteros
A first proof of knot localization for polymers in a nanochannel
Journal of Physics A: Mathematical and Theoretical 57 (2024) 38LT01
Link | Show/Hide Abstract
Based on polymer scaling theory and numerical evidence, Orlandini, Tesi, Janse van Rensburg and Whittington conjectured in 1996 that the limiting entropy of knot-type $K$ lattice polygons is the same as that for unknot polygons, and that the entropic critical exponent increases by one for each prime knot in the knot decomposition of $K$. This Knot Entropy (KE) conjecture is consistent with the idea that for unconfined polymers, knots occur in a localized way (the knotted part is relatively small compared to polymer length). For full confinement (to a sphere or box), numerical evidence suggests that knots are much less localized. Numerical evidence for nanochannel or tube confinement is mixed, depending on how the size of a knot is measured. Here we outline the proof that the KE conjecture holds for polygons in the $\infty\times 2\times 1$ lattice tube and show that knotting is localized when a connected-sum measure of knot size is used. Similar results are established for linked polygons. This is the first model for which the knot entropy conjecture has been proved.
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 (2023) 155003
Link | 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.
Nicholas R. Beaton and Mark Holmes
The mean square displacement of random walk on the Manhattan lattice
Statistics & Probability Letters 193 (2023) 109706
Link | arXiv | Show/Hide Abstract
We give an explicit formula for the mean square displacement of the random walk on the $d$-dimensional Manhattan lattice after $n$ steps, for all $n$ and all dimensions $d\ge 2$.
Nicholas R. Beaton
Walks obeying two-step rules on the square lattice: full, half and quarter planes
Electronic Journal of Combinatorics 29 (2022) P1.14
Link | arXiv | Code | Show/Hide Abstract
We consider walks on the edges of the square lattice $\mathbb Z^2$ which obey two-step rules, which allow (or forbid) steps in a given direction to be followed by steps in another direction. We classify these rules according to a number of criteria, and show how these properties affect their generating functions, asymptotic enumerations and limiting shapes, on the full lattice as well as the upper half plane.
For walks in the quarter plane, we only make a few tentative first steps. We propose candidates for the group of a model, analogous to the group of a regular short-step quarter plane model, and investigate which models have finite versus infinite groups. We demonstrate that the orbit sum method used to solve a number of the original models can be made to work for some models here, producing a D-finite solution. We also generate short series for all models and guess differential or algebraic equations where possible. In doing so, we find that there are possibilities here which do not occur for the regular short-step models, including cases with algebraic or D-finite generating functions but infinite groups, as well as models with non-D-finite generating functions but finite groups.
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, Geoffrey R. Grimmett, and Mark Holmes
Alignment percolation
Mathematical Physics, Analysis and Geometry 24 (2021) #3.
Link | arXiv | Show/Hide Abstract
The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in $d \ge 2$ dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on ${\mathbb Z}^d$ with parameter $p\in (0,1]$. For each occupied site $v$, and for each of the $2d$ possible coordinate directions, declare the entire line segment from $v$ to the next occupied site in the given direction to be either blue or not blue according to a given stochastic rule. In the one-choice model, each occupied site declares one of its $2d$ incident segments to be blue. In the independent model, the states of different line segments are independent.
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 | 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.
Nicholas R. Beaton, Anthony J. Guttmann, and Iwan Jensen
Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents
Journal of Physics A: Mathematical and Theoretical 53 (2020) 165002.
Link | arXiv | Show/Hide Abstract
We investigate, by series methods, the behaviour of interacting self-avoiding walks (ISAWs) on the honeycomb lattice and on the square lattice. This is the first such investigation of ISAWs on the honeycomb lattice. We have generated data for ISAWs up to 75 steps on this lattice, and 55 steps on the square lattice. For the hexagonal lattice we find the $\theta$-point to be at $u_\mathrm{c} = 2.767 \pm 0.002.$ The honeycomb lattice is unique among the regular two-dimensional lattices in that the exact growth constant is known for non-interacting walks, and is $\sqrt{2+\sqrt{2}}$, while for half-plane walks interacting with a surface, the critical fugacity, again for the honeycomb lattice, is $1+\sqrt{2}$. We could not help but notice that $\sqrt{2+4\sqrt{2}} = 2.767\ldots .$ We discuss the difficulties of trying to prove, or disprove, this possibility.
For square lattice ISAWs we find $u_\mathrm{c}=1.9474 \pm 0.001,$ which is consistent with the best Monte Carlo analysis.
We also study bridges and terminally-attached walks (TAWs) on the square lattice at the $\theta$-point. We estimate the exponents to be $\gamma_b=0.00 \pm 0.03,$ and $\gamma_1=0.55 \pm 0.03$ respectively. The latter result is consistent with the prediction $\gamma_1(\theta) = \nu =\frac47$, albeit for a modified version of the problem, while the former estimate appears to be new.
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.
Nicholas R. Beaton, Aleksander L. Owczarek, and Andrew Rechnitzer
Exact solution of some quarter plane walks with interacting boundaries
Electronic Journal of Combinatorics 26 (2019) P3.53.
Link | 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.
Nicholas R. Beaton, Mathilde Bouvel, Veronica Guerrini, and Simone Rinaldi
Slicings of parallelogram polyominoes: Catalan, Schröder, Baxter, and other sequences
Electronic Journal of Combinatorics 26 (2019) P3.13.
Link | arXiv | Show/Hide Abstract
We provide a new succession rule (i.e. generating tree) associated with Schröder numbers,
that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we present an underpinned conjecture that they are algebraic for any $m$.
Nicholas R. Beaton, Mathilde Bouvel, Veronica Guerrini, and Simone Rinaldi
Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers
Theoretical Computer Science 777 (2019) 69-92.
Link | arXiv | Show/Hide Abstract
The first problem addressed by this article is the enumeration of some families of pattern-avoiding inversion sequences. We solve some enumerative conjectures left open by the foundational work on the topics by Corteel et al., some of these being also solved independently by Kim and Lin. The strength of our approach is its robustness: we enumerate four families $F_1\subset F_2 \subset F_3 \subset F_4$ of pattern-avoiding inversion sequences ordered by inclusion using the same approach. More precisely, we provide a generating tree (with associated succession rule) for each family $F_i$ which generalizes the one for the family $F_{i−1}$.
The second topic of the paper is the enumeration of a fifth family $F_5$ of pattern-avoiding inversion sequences (containing $F_4$). This enumeration is also solved via a succession rule, which however does not generalize the one for $F_4$. The associated enumeration sequence, which we call of powered Catalan numbers, is quite intringuing, and further investigated. We provide two different succession rules for it, denoted $\Omega_{pCat}$ and $\Omega_{steady}$, and show that they define two types of families enumerated by powered Catalan numbers. Among such families, we introduce the steady paths, which are naturally associated with $\Omega_{steady}$. They allow to bridge the gap between the two types of families enumerated by powered Catalan numbers: indeed, we provide a size-preserving bijection between steady paths and valley-marked Dyck paths (which are naturally associated to $\Omega_{pCat}$).
Along the way, we provide several nice connections to families of permutations defined by the avoidance of vincular patterns, and some enumerative conjectures.
Nicholas R. Beaton, Jeremy W. Eng, and Christine E. Soteros
Knotting statistics for polygons in lattice tubes
Journal of Physics A: Mathematical and Theoretical 52 (2019) 144003.
Link | arXiv | PDF File | Show/Hide Abstract
We study several related models of self-avoiding polygons in a tubular subgraph of the simple cubic lattice, with a particular interest in the asymptotics of the knotting statistics. Polygons in a tube can be characterised by a finite transfer matrix, and this allows for the derivation of pattern theorems, calculation of growth rates and exact enumeration. We also develop a static Monte Carlo method which allows us to sample polygons of a given size directly from a chosen Boltzmann distribution.
Using these methods we accurately estimate the growth rates of unknotted polygons in the $2\times1\times\infty$ and $3\times1\times\infty$ tubes, and confirm that these are the same for any fixed knot-type $K$. We also confirm that the entropic exponent for unknots is the same as that of all polygons, and that the exponent for fixed knot-type $K$ depends only on the number of prime factors in the knot decomposition of $K$. For the simplest knot-types, this leads to a good approximation for the polygon size at which the probability of the given knot-type is maximized, and in some cases we are able to sample sufficiently long polygons to observe this numerically.
Alireza Montazeri, Nicholas R. Beaton, and Dwight Makaroff
LRU-2 vs. 2-LRU: An Analytical Study
Proceedings of the IEEE 43rd Conference on Local Computer Networks (LCN), 2018.
Link | PDF file | Show/Hide Abstract | Copyright Notice
Hierarchical caching enables users to obtain content from one of (possibly) many caches between an edge router cache and an origin server, reducing latency/overall network traffic. This can be used in many network architectures, including P2P networks and Content Distribution Networks. Of particular interest are Information Centric Networks (ICNs), which decouple content identifiers from specific network hosts and explicitly consider universal caching (collaboration between network routers) as a desirable feature.
Performance analysis of a large-scale network of caches requires accurate mathematical models for various caching replacement algorithms. There is no previous study that models LRU-$k$. This caching replacement algorithm is important since its principle is the basis of recent caching replacement algorithms such as $k$-LRU that outperform LRU in many situations. The main contribution in this paper is modelling LRU-2, using Che's approximation, as a specific case of LRU-$k$ for $k=2$. We also extend our model to a network of LRU-2 caches. The model is validated analytically and with simulation. The experiments show that the proposed model approximates the LRU-2 algorithm accurately. LRU-2 and 2-LRU are also compared analytically and with simulations as the second contribution.
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Nicholas R. Beaton, Jeremy W. Eng, Kai Ishihara, Koya Shimokawa, and Christine E. Soteros
Characterising knotting properties of polymers in nanochannels
Soft Matter 14 (2018), 5775-5785
Link | arXiv | PDF file and Electronic Supplementary Information | Show/Hide Abstract
Using a lattice model of polymers in a tube, we define one way to characterise different configurations of a given knot as either “local” or “non-local”, based on a standard approach for measuring the “size” of a knot within a knotted polymer chain. The method involves associating knot-types to subarcs of the chain, and then identifying a knotted subarc with minimal arclength; this arclength is then the knot-size. If the resulting knot-size is small relative to the whole length of the chain, then the knot is considered to be localised or “local”; otherwise, it is “non-local”. Using this definition, we establish that all but exponentially few sufficiently long self-avoiding polygons (closed chains) in a tubular sublattice of the simple cubic lattice are “non-locally” knotted. This is shown to also hold for the case when the same polygons are subject to an external tensile force, as well as in the extreme case when they are as compact as possible (no empty lattice sites). We also provide numerical evidence for small tube sizes that at equilibrium non-local knotting is more likely than local knotting, regardless of the strength of the stretching or compressing force. The relevance of these results to other models and recent experiments involving DNA knots is also discussed.
Nicholas R. Beaton, Andrew R. Conway, and Anthony J. Guttmann
On consecutive pattern-avoiding permutations of length 4, 5 and beyond
Discrete Mathematics and Theoretical Computer Science 19 -- Special Issue for Permutation Patterns 2016 (2018), Article #8
Link | arXiv | PDF file | Show/Hide Abstract
We review and extend what is known about the generating functions for consecutive pattern avoiding permutations of length 4, 5 and beyond, and their asymptotic behaviour. There are respectively, seven length-4 and twenty-five length-5 consecutive-Wilf classes. D-finite differential equations are known for the reciprocal of the exponential generating functions for four of the length-4 and eight of the length-5 classes. We give the solutions of some of these ODEs. An unsolved functional equation is known for one more class of length-4, length-5 and beyond. We give the solution of this functional equation, and use it to show that the solution is not D-finite.
For three further length-5 c-Wilf classes we give recurrences for two and a differential-functional equation for a third. For a fourth class we find a new algebraic solution.
We give a polynomial-time algorithm to generate the coefficients of the generating functions which is faster than existing algorithms, and use this to (a) calculate the asymptotics for all classes of length 4 and length 5 to significantly greater precision than previously, and (b) use these extended series to search, unsuccessfully, for D-finite solutions for the unsolved classes, leading us to conjecture that the solutions are not D-finite. We have also searched, unsuccessfully, for differentially algebraic solutions.
Nicholas R. Beaton and E.J. Janse van Rensburg
Partition function zeros of adsorbing Dyck paths
Journal of Physics A: Mathematical and Theoretical 51 (2018), 114002
Link | arXiv | PDF file | Show/Hide Abstract
The zeros of the size-$n$ partition functions for a statistical mechanical model can be used to help understand the critical behaviour of the model as $n\to\infty$. Here we use weighted Dyck paths as a simple model of two-dimensional polymer adsorption, and study the behaviour of the partition function zeros, particularly in the thermodynamic limit. The exact solvability of the model allows for a precise calculation of the locus of the zeros and the way in which an edge-singularity on the positive real axis is formed.
Nicholas R. Beaton
Adsorbing staircase polygons subject to a force
Journal of Physics A: Mathematical and Theoretical 50 (2017), 494001
Link | arXiv | PDF file | Show/Hide Abstract
We study several models of staircase polygons on the $45^\circ$ rotated square lattice, which interact with an impenetrable surface while also being pushed towards or pulled away from the surface by a force. The surface interaction is governed by a fugacity $a$ and the force by a fugacity $y$. Staircase polygons are simplifications of more general self-avoiding polygons, a well-studied model of interacting ring polymers. For this simplified case we are able to exactly determine the limiting free energy in the full $a$-$y$ plane, and demonstrate that staircase polygons exhibit four different phases, including a "mixed" adsorbed-ballistic phase.
Nicholas R. Beaton, Jeremy W. Eng, and Christine E. Soteros
Polygons in restricted geometries subjected to infinite forces
Journal of Physics A: Mathematical and Theoretical 49 (2016), 424002
Link | arXiv | PDF file | Show/Hide Abstract
We consider self-avoiding polygons in a restricted geometry, namely an infinite $L\times M$ tube in $\mathbb Z^3$. These polygons are subjected to a force $f$, parallel to the infinite axis of the tube. When $f>0$ the force stretches the polygons, while when $f\lt0$ the force is compressive. We obtain and prove the asymptotic form of the free energy in both limits $f\to\pm\infty$. We conjecture that the $f\to-\infty$ asymptote is the same as the limiting free energy of "Hamiltonian" polygons, polygons which visit every vertex in a $L\times M\times N$ box. We investigate such polygons, and in particular use a transfer-matrix methodology to establish that the conjecture is true for some small tube sizes.
$\rightarrow$ See also the version which appeared in
Proceedings of the 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016, Vancouver, Canada), 119-130
Link | PDF file | Show/Hide Abstract
We consider self-avoiding polygons in a restricted geometry, namely an infinite $L\times M$ tube in $\mathbb{Z}^3$. These polygons are subjected to a force $f$, parallel to the infinite axis of the tube. When $f>0$ the force stretches the polygons, while when $f<0$ the force is compressive. In this extended abstract we obtain and prove the asymptotic form of the free energy in the limit $f\to-\infty$. We conjecture that the $f\to-\infty$ asymptote is the same as the free energy of Hamiltonian polygons, which visit every vertex in a $L\times M\times N$ box.
Nicholas R. Beaton, Anthony J. Guttmann, Iwan Jensen, and Gregory F. Lawler
Compressed self-avoiding walks, bridges and polygons
Journal of Physics A: Mathematical and Theoretical 48 (2015), 454001
Link | arXiv | PDF file | Show/Hide Abstract
We study various self-avoiding walks (SAWs) which are constrained to lie in the upper half-plane and are subjected to a compressive force. This force is applied to the vertex or vertices of the walk located at the maximum distance above the boundary of the half-space. In the case of bridges, this is the unique end-point. In the case of SAWs or self-avoiding polygons, this corresponds to all vertices of maximal height. We first use the conjectured relation with the Schramm-Loewner evolution to predict the form of the partition function including the values of the exponents, and then we use series analysis to test these predictions.
Nicholas R. Beaton and Gerasim K. Iliev
Two-sided prudent walks: A solvable non-directed model of polymer adsorption
Journal of Statistical Mechanics: Theory and Experiment (2015), P09014
Link | arXiv | PDF file | Show/Hide Abstract
Prudent walks are self-avoiding walks which cannot step towards an already occupied vertex. We introduce a new model of adsorbing prudent walks on the square lattice, which start on an impenetrable surface and accrue a fugacity $a$ with each step along the surface. These are different to other exactly solved models of polymer adsorption, like Dyck paths, Motzkin paths and partially-directed walks, in that they are not trivially directed - they are able to step in all lattice directions. We calculate the generating functions, free energies and surface densities for this model and observe a first-order adsorption transition at the critical value of the surface interaction.
Nicholas R. Beaton
The critical pulling force for self-avoiding walks
Journal of Physics A: Mathematical and Theoretical 48 (2015), 16FT03
Link | arXiv | PDF file | Show/Hide Abstract
Self-avoiding walks are a simple and well-known model of long, flexible polymers in a good solvent. Polymers being pulled away from a surface by an external agent can be modelled with self-avoiding walks in a half-space, with a Boltzmann weight $y = e^f$ associated with the pulling force. This model is known to have a critical point at a certain value $y_c$ of this Boltzmann weight, which is the location of a transition between the so-called free and ballistic phases. The value $y_c=1$ has been conjectured by several authors using numerical estimates. We provide a relatively simple proof of this result, and show that further properties of the free energy of this system can be determined by re-interpreting existing results about the two-point function of self-avoiding walks.
Axel Bacher and Nicholas R. Beaton
Weakly prudent self-avoiding bridges
Proceedings of the 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014, Chicago, USA), 827-838
Link | PDF file | Show/Hide Abstract
We define and enumerate a new class of self-avoiding walks on the square lattice, which we call weakly prudent bridges. Their definition is inspired by two previously-considered classes of self-avoiding walks, and can be viewed as a combination of those two models. We consider several methods for recursively generating these objects, each with its own advantages and disadvantages, and use these methods to solve the generating function, obtain very long series, and randomly generate walks of arbitrary size. We find that the growth constant of these walks is approximately 2.58, which is larger than that of any previously-solved class of self-avoiding walks.
Nicholas R. Beaton, Mireille Bousquet-Mélou, Jan de Gier, Hugo Duminil-Copin, and Anthony J. Guttmann
The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is $1+\sqrt{2}$
Communications in Mathematical Physics 326 (2014), 727-754
Link | arXiv | PDF file | Show/Hide Abstract
In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis, made in 1982, that the growth constant of self-avoiding walks on the hexagonal (a.k.a. honeycomb) lattice is $\mu = \sqrt{2+\sqrt{2}}$. A key identity used in that proof was later generalised by Smirnov so as to apply to a general $O(n)$ loop model with $n\in[−2,2]$ (the case $n=0$ corresponding to self-avoiding walks).
We modify this model by restricting to a half-plane and introducing a surface fugacity y associated with boundary sites (also called surface sites), and obtain a generalisation of Smirnov’s identity. The critical value of the surface fugacity was conjectured by Batchelor and Yung in 1995 to be $y_c=1+2/\sqrt{2-n}$. This value plays a crucial role in our generalized identity, just as the value of the growth constant did in Smirnov’s identity.
For the case $n=0$, corresponding to self-avoiding walks interacting with a surface, we prove the conjectured value of the critical surface fugacity. A crucial part of the proof involves demonstrating that the generating function of self-avoiding bridges of height $T$, taken at its critical point $1/\mu$, tends to 0 as $T$ increases, as predicted from SLE theory.
Nicholas R. Beaton
The critical surface fugacity for self-avoiding walks on a rotated honeycomb lattice
Journal of Physics A: Mathematical and Theoretical 47 (2014), 075003+
Link | arXiv | PDF file | Show/Hide Abstract
In a recent paper by Beaton et al, it was proved that a model of self-avoiding walks on the honeycomb lattice, interacting with an impenetrable surface, undergoes an adsorption phase transition when the surface fugacity is $1+\sqrt{2}$. Their proof used a generalization of an identity obtained by Duminil-Copin and Smirnov, and confirmed a conjecture of Batchelor and Yung. We consider a similar model of self-avoiding walk adsorption on the honeycomb lattice, but with the lattice rotated by $\pi/2$. For this model there also exists a conjecture for the critical surface fugacity, made in 1998 by Batchelor, Bennett-Wood and Owczarek. Using similar methods to Beaton et al, we prove that this is indeed the critical fugacity.
$\rightarrow$ See also the version which appeared in
Proceedings of the 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013, Paris, France), 665-676
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In a recent paper with Bousquet-Mélou, de Gier, Duminil-Copin and Guttmann (2012), we proved that a model of self-avoiding walks on the honeycomb lattice, interacting with an impenetrable surface, undergoes an adsorption phase transition when the surface fugacity is $1+\sqrt{2}$. Our proof used a generalisation of an identity obtained by Duminil-Copin and Smirnov (2012), and confirmed a conjecture of Batchelor and Yung (1995). Here we consider a similar model of self-avoiding walk adsorption on the honeycomb lattice, but with the impenetrable surface placed at a right angle to the previous orientation. For this model there also exists a conjecture for the critical surface fugacity, made by Batchelor, Bennett-Wood and Owczarek (1998). We adapt the methods of the earlier paper to this setting in order to prove the critical surface fugacity, but have to deal with several subtle complications which arise. This article is an abbreviated version of a paper of the same title, currently being prepared for submission.
Nicholas R. Beaton, Anthony J. Guttmann, and Iwan Jensen
Two-dimensional self-avoiding walks and polymer adsorption: Critical fugacity estimates
Journal of Physics A: Mathematical and Theoretical 45 (2012), 055208+
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Recently Beaton, de Gier and Guttmann proved a conjecture of Batchelor and Yung that the critical fugacity of self-avoiding walks (SAW) interacting with (alternate) sites on the surface of the honeycomb lattice is $1+\sqrt{2}$. A key identity used in that proof depends on the existence of a parafermionic observable for SAW interacting with a surface on the honeycomb lattice. Despite the absence of a corresponding observable for SAW on the square and triangular lattices, we show that in the limit of large lattices, some of the consequences observed for the honeycomb lattice persist irrespective of lattice. This permits the accurate estimation of the critical fugacity for the corresponding problem for the square and triangular lattices. We consider both edge and site weighting, and results of unprecedented precision are achieved. We also prove the corresponding result for the edge-weighted case for the honeycomb lattice.
Nicholas R. Beaton, Anthony J. Guttmann, and Iwan Jensen
A numerical adaptation of SAW identities from the honeycomb to other 2D lattices
Journal of Physics A: Mathematical and Theoretical 45 (2012), 035201+
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Recently, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis that the connective constant of self-avoiding walks (SAWs) on the honeycomb lattice is $\sqrt{2+\sqrt{2}}$. A key identity used in that proof depends on the existence of a parafermionic observable for SAWs on the honeycomb lattice. Despite the absence of a corresponding observable for SAWs on the square and triangular lattices, we show that in the limit of large lattices, some of the consequences observed on the honeycomb lattice persist on other lattices. This permits the accurate estimation, though not an exact evaluation, of certain critical amplitudes, as well as critical points, for these lattices. For the honeycomb lattice, an exact amplitude for loops is proved.
Nicholas R. Beaton, Philippe Flajolet, Timothy M. Garoni, and Anthony J. Guttmann
Some new self-avoiding walk and polygon models
Fundamenta Informaticae 117 (2012), 19-33
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We study the behaviour of prudent, perimeter and quasi-prudent self-avoiding walks and polygons in both two and three dimensions, as well as some solvable subsets. Our analysis combines exact solutions of some simpler cases, careful asymptotic analysis of functional equations which can be obtained in more complicated cases and extensive numerical studies based on exact series expansions for less tractable cases, augmented by long Monte Carlo runs in some cases.
Nicholas R. Beaton, Philippe Flajolet, and Anthony J. Guttmann
The enumeration of prudent polygons by area and its unusual asymptotics
Journal of Combinatorial Theory, Series A 118 (2011), 2261-2290
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Prudent walks are special self-avoiding walks that never take a step towards an already occupied site, and $k$-sided prudent walks (with $k=1,2,3,4$) are, in essence, only allowed to grow along $k$ directions. Prudent polygons are prudent walks that return to a point adjacent to their starting point. Prudent walks and polygons have recently been enumerated by length and perimeter by Bousquet-Mélou and Schwerdtfeger. We consider the enumeration of prudent polygons by area. For the 3-sided variety, we find that the generating function is expressed in terms of a $q$-hypergeometric function, with an accumulation of poles towards the dominant singularity. This expression reveals an unusual asymptotic structure of the number of polygons of area $n$, where the critical exponent is the transcendental number $\log_{2}3$ and the amplitude involves tiny oscillations. Based on numerical data, we also expect similar phenomena to occur for 4-sided polygons. The asymptotic methodology involves an original combination of Mellin transform techniques and singularity analysis, which is of potential interest in a number of other asymptotic enumeration problems.
Nicholas R. Beaton, Filippo Disanto, Anthony J. Guttmann, and Simone Rinaldi
On the enumeration of column-convex permutominoes
Proceedings of the 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011, Reykjavik, Iceland), 111-122
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We study the enumeration of column-convex permutominoes, i.e. column-convex polyominoes defined by a pair of permutations. We provide a direct recursive construction for the column-convex permutominoes of a given size, based on the application of the ECO method and generating trees, which leads to a functional equation. Then we obtain some upper and lower bounds for the number of column-convex permutominoes, and conjecture its asymptotic behavior using numerical analysis.
Nicholas R. Beaton, Philippe Flajolet, and Anthony J. Guttmann
The unusual asymptotics of three-sided prudent polygons
Journal of Physics A: Mathematical and Theoretical 43 (2010), 342001+
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We have studied the area-generating function of prudent polygons on the square lattice. Exact solutions are obtained for the generating function of two-sided and three-sided prudent polygons, and a functional equation is found for four-sided prudent polygons. This is used to generate series coefficients in polynomial time, and these are analysed to determine the asymptotics numerically. A careful asymptotic analysis of the three-sided polygons produces a most surprising result. A transcendental critical exponent is found, and the leading amplitude is not quite a constant, but is a constant plus a small oscillatory component with an amplitude approximately $10^{−8}$ times that of the leading amplitude. This effect cannot be seen by any standard numerical analysis, but it may be present in other models. If so, it changes our whole view of the asymptotic behaviour of lattice models.
PhD Thesis
Combinatorics of Lattice Paths and Polygons
Department of Mathematics and Statistics
The University of Melbourne, 2012
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We consider the enumeration of self-avoiding walks and polygons on regular lattices. Such objects are connected with many other problems in combinatorics, as well as in fields as diverse as physics and chemistry. We examine the general models of walks and polygons and methods we can use to study them; subclasses whose properties enable a rather deeper analysis; and extensions of these models which allow us to model physical phenomena like polymer collapse and adsorption.
While the general models of self-avoiding walks and polygons are certainly not considered to be ‘solved’, recently a great deal of progress has been made in developing new methods for studying these objects and proving rigorous results about their enumerative properties, particularly on the honeycomb lattice. We consider these recent results and show that in some cases they can be extended or generalised so as to enable further proofs, conjectures and estimates. In particular, we find that properties shared by all two-dimensional lattices allow us to develop new methods for estimating the growth constants and certain amplitudes for the square and triangular lattices.
The subclasses of self-avoiding walks and polygons that we consider are typically defined by imposing restrictions on the way in which a walk or polygon can be constructed. Ideally the restrictions should be as weak as possible, so as to result in a model that closely resembles the unrestricted case, while still enabling some manner of simple recursive construction. These recursions can sometimes lead to solutions for generating functions or other quantities of interest. Some of the models we consider display quite unusual asymptotic properties despite the relatively simple restrictions which lead to their construction.
We approach the modelling of polymer adsorption in several ways. Firstly, we adapt some of the new methods for studying self-avoiding walks on the honeycomb lattice to account for interactions with an impenetrable surface. In this way we are able to prove the exact value of the critical surface fugacity for adsorbing walks, confirming an existing conjecture. Then, we show that some key identities for the honeycomb lattice model lead to a new method for estimating the critical surface fugacities for adsorption models on the square and triangular lattices. Many of the estimates we obtain in this way are new; for the cases where previous estimates did already exist, our results are several orders of magnitude more precise. Finally, we define some new solvable models of polymer adsorption which generalise existing models, and find that some of these models exhibit interesting and unexpected critical behaviour.
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