Anthony Guttmann

• M.Sc (Melbourne) PhD (UNSW)
• Professor at University of Melbourne

We derive a simple functional equation with two catalytic variables characterising the generating function of 3-stack-sortable permutations. Using this functional equation, we extend the 174-term series to 1000 terms. From this series, we conjecture that the generating function behaves as $$W(t) \sim C_0(1-\mu_3 t)^\alpha \cdot \log^\beta(1-\mu_3 t...

We show how the theory of the critical behaviour of d-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain in a bridge configuration. We further define multi-bridge networks, where several vertices are in local bridge...

We show how the theory of the critical behaviour of $d$-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain in a \emph{bridge} configuration. We further define multi-bridge networks, where several vertices are in loca...

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...

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...

We consider self-avoiding lattice polygons, in the square and cubic lattices, as a model of a ring polymer adsorbed at a surface and either being desorbed by the action of a force, or pushed towards the surface. We show that, when there is no interaction with the surface, then the response of the polygon to the applied force is identical (in the th...

We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in $14$ further terms of the generating function, which is now known for all patterns of length $\le 50$. We re-analyse the generating function and find additional evidence for our earlier conclusion that unlike other classical length-$4$ pattern-avoidi...

Inspired by the paper of Bonichon, Bousquet-M\'elou, Dorbec and Pennarun, we give a system of functional equations which characterise the ordinary generating function, $U(x),$ for the number of planar Eulerian orientations counted by edges. We also characterise the ogf $A(x)$, for 4-valent planar Eulerian orientations counted by vertices in a simil...

We have developed polynomial-time algorithms to generate terms of the cogrowth series for groups [Formula: see text], the lamplighter group, [Formula: see text] and the Brin–Navas group [Formula: see text]. We have also given an improved algorithm for the coefficients of Thompson’s group [Formula: see text], giving 32 terms of the cogrowth series....

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 generatin...

We consider self-avoiding lattice polygons, in the hypercubic lattice, as a model of a ring polymer adsorbed at a surface and either being desorbed by the action of a force, or pushed towards the surface. We show that, when there is no interaction with the surface, then the response of the polygon to the applied force is identical (in the thermodyn...

We study the class of non-holonomic power series with integer coefficients that reduce, modulo primes, or powers of primes, to algebraic functions. In particular we try to determine whether the susceptibility of the square-lattice Ising model belongs to this class, and more broadly whether the susceptibility is a solution of a differentially algebr...

The growth constant for two-dimensional self-avoiding walks on the honeycomb lattice was conjectured by Nienhuis in 1982, and since that time the corresponding results for the square and triangular lattices have been sought. For the square lattice, a possible conjecture was advanced by one of us (AJG) more than 20 years ago, based on the six signif...

Given the first 20-100 coefficients of a typical generating function of the type that arises in many problems of statistical mechanics or enumerative combinatorics, we show that the method of differential approximants performs surprisingly well in predicting (approximately) subsequent coefficients. These can then be used by the ratio method to obta...

We study the class of non-holonomic power series with integer coefficients that reduce, modulo primes, or powers of primes, to algebraic functions. In particular we try to determine whether the susceptibility of the square-lattice Ising model belongs to this class, and more broadly whether the susceptibility is a solution of a differentially algebr...

We study a one-parameter family ($\ell=1,2,3,\ldots$) of configurations that are square-ice analogues of plane partitions. Using an algorithm due to Bratley and McKay, we carry out exact enumerations in order to study their asymptotic behaviour and establish, via Monte Carlo simulations as well as explicit bounds, that the asymptotic behaviour is s...

We study terminally attached self-avoiding walks and bridges on the simple-cubic lattice, both by series analysis and Monte Carlo methods. We provide strong numerical evidence supporting a scaling relation between self-avoiding walks, bridges, and terminally attached self-avoiding walks, and posit that a corresponding amplitude ratio is a universal...

We have studied the problem of the number of permutations that can be sorted by two stacks in series. We do this by first counting all such permutations of length less than 20 exactly, then using a numerical technique to obtain eleven further coefficients approximately. Analysing these coefficients by a variety of methods we conclude that the OGF b...

We study the full susceptibility of the Ising model modulo powers of primes. We find exact functional equations for the full susceptibility modulo these primes. Revisiting some lesser-known results on discrete finite automata, we show that these results can be seen as a consequence of the fact that, modulo 2r , one cannot distinguish the full susce...

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...

Recently Albert and Bousquet-M\'elou \cite{AB15} obtained the solution to the long-standing problem of the number of permutations sortable by two stacks in parallel (tsip). Their solution was expressed in terms of functional equations. We show that the equally long-standing problem of the number of permutations sortable by a deque can be simply rel...

We give an improved algorithm for counting the number of 1324-avoiding permutations, resulting in 5 further terms of the generating function. We analyse the known coefficients and find compelling evidence that unlike other classical length-4 pattern-avoiding permutations, the generating function in this case does not have an algebraic singularity....

Existing methods of series analysis are largely designed to analyse the structure of algebraic singularities. Functions with such singularities have their $n^{th}$ coefficient behaving asymptotically as $A \cdot \mu^n \cdot n^g.$ Recently, a number of problems in statistical mechanics and combinatorics have been encountered in which the coefficient...

We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in 5 further terms of the generating function. We analyse the known coefficients and find compelling evidence that unlike other classical length-4 pattern-avoiding permutations, the generating function in this case does not have an algebraic singularity...

We consider a self-avoiding walk model of polymer adsorption where the adsorbed polymer can be desorbed by the application of a force, concentrating on the case of the square lattice. Using series analysis methods we investigate the behaviour of the free energy of the system when there is an attractive potential $\epsilon$ with the surface and a fo...

It is widely believed that the scaling limit of self-avoiding walks (SAWs) at the critical temperature is (i) conformally invariant, and (ii) describable by Schramm-Loewner Evolution (SLE) with parameter $\kappa = 8/3.$ We consider SAWs in a rectangle, which originate at its centre and end when they reach the boundary. We assume that the scaling li...

We show that the integral $J(t) = (1/\pi^3) \int_0^\pi \int_0^\pi \int_0^\pi dx dy dz \log(t - \cos{x} - \cos{y} - \cos{z} + \cos{x}\cos{y}\cos{z})$, can be expressed in terms of ${_5F_4}$ hypergeometric functions. The integral arises in the solution by Baxter and Bazhanov of the free-energy of the $sl(n)$ Potts model, which includes the term $J(2)...

This is a rather personal review of the problem of self-avoiding walks and polygons. After defining the problem, and outlining what is known rigorously and what is merely conjectured, I highlight the major outstanding problems. I then give several applications in which the I have been involved. These include a study of surface adsorption of polymer...

Fa Yueh (Fred) Wu was born on 5 January 1932 in Nanking (now known as Nanjing), China, the capital of the Nationalist government. Wu began kindergarten in 1937 in a comfortable setting, as his father held a relatively high government position. But the Sino–Japanese war broke out in July 1937, and Nanking fell to Japanese hands in November. Fleeing...

A celebrated problem in numerical analysis is to consider Brownian motion originating at the centre of a $10 \times 1$ rectangle, and to evaluate the ratio of probabilities of a Brownian path hitting the short ends of the rectangle before hitting one of the long sides. For Brownian motion this probability can be calculated exactly \cite{BLWW04}. He...

We define the notion of a spanning tree generating function (STGF) ∑anzⁿ, which gives the spanning tree constant when evaluated at z = 1, and gives the lattice Green function (LGF) when differentiated. By making use of known results for logarithmic Mahler measures of certain Laurent polynomials, and proving new results, we express the STGFs as hype...

Using an off-critical deformation of the identity of Duminil-Copin and Smirnov, we prove a relationship between half-plane surface critical exponents $\gamma_1$ and $\gamma_{11}$ as well as wedge critical exponents $\gamma_2(\alpha)$ and $\gamma_{21}(\alpha)$ and the exponent characterising the winding angle distribution of the O($n$) model in the...

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 s...

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 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 SAW...

International audience We study the enumeration of \emphcolumn-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....

Recently Beaton, de Gier and Guttmann proved a conjecture of Batchelor and Yung that the critical fugacity of self-avoiding walks 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 self-avoiding walks interacting wit...

Recently, Duminil-Copin and Smirnov proved a long-standing conjecture by Nienhuis that the connective constant of self-avoiding walks 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 self-avoiding walks on the honeycomb lattice. Despite the absence of a cor...

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]...

Recently Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis, made in 1982, that the connective constant of self-avoiding walks on the honeycomb lattice is $\sqrt{2+\sqrt{2}}.$ A key identity used in that proof was later generalised by Smirnov so as to apply to a general O(n) model with $n\in [-2,2]$. We modify this model by res...

We have dramatically extended the zero field susceptibility series at both high and low temperature of the Ising model on the triangular and honeycomb lattices, and used these data and newly available further terms for the square lattice to calculate a number of terms in the scaling function expansion around both the ferromagnetic and, for the squa...

Prudent walks are special self-avoiding walks that never take a step towards an already occupied site, and \emph{$k$-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 been prev...

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...

We give a systematic treatment of lattice Green's functions (LGF) on the d-dimensional diamond, simple cubic, body-centred cubic and face-centred cubic lattices for arbitrary dimensionality d ≥ 2 for the first three lattices, and for 2 ≤ d ≤ 5 for the hyper-fcc lattice. We show that there is a close connection between the LGF of the d-dimensional h...

We obtain in exact arithmetic the order 24 linear differential operator L24 and the right-hand side E(5) of the inhomogeneous equation L24(Φ(5)) = E(5), where is a linear combination of n-particle contributions to the susceptibility of the square lattice Ising model. In Bostan et al (2009 J. Phys. A: Math. Theor. 42 275209), the operator L24 (modul...

We derive a Toda-type recurrence relation, in both high- and low-temperature regimes, for the λ-extended diagonal correlation functions C(N,N;λ) of the two-dimensional Ising model, using an earlier connection between diagonal form factor expansions and tau-functions within Painlevé VI (PVI) theory, originally discovered by Jimbo and Miwa. This grea...

Column-convex polyominoes were introduced in 1950's by Temperley, a mathematical physicist working on "lattice gases". By now, column-convex polyominoes are a popular and well-understood model. There exist several generalizations of column-convex polyominoes; an example is a model called multi-directed animals. In this paper, we introduce a new seq...

Column-convex polygons were first counted by area several decades ago, and the result was found to be a simple, rational, generating func-tion. In this work we generalize that result. Let a p-column polyomino be a polyomino whose columns can have 1, 2, . . . , p connected components. Then column-convex polygons are equivalent to 1-convex polyominoe...

By making the connection between four-dimensional lattice Green functions (LGFs) and Picard–Fuchs ordinary differential equations of Calabi–Yau manifolds, we have given explicit forms for the coefficients of the four-dimensional LGFs on the simple-cubic and body-centred cubic lattices, in terms of finite sums of products of binomial coefficients, a...

We consider the Fuchsian linear differential equation obtained (modulo a prime) for $\tilde{\chi}^{(5)}$, the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particula...

In this book we will primarily be concerned with the properties and applications of self-avoiding polygons (SAP). Two closely related problems are those of polyomi-noes, and the much broader one of tilings. We will describe and discuss polyominoes and, in the context of a discussion of SAP, will briefly mention relevant aspects of the subject of ti...

In this chapter we will be considering the effect of confining polygons to lie in a bounded geometry. This has already been briefly discussed in Chapters 2 and 3, but here we give many more results. The simplest, non-trivial case is that of SAP on the two-dimensional square lattice Z2, confined between two parallel lines, say x = 0 and x = w. This...

Since a connection was made in the 19th Century between increase of entropy and earlier expressions of the Second Law of Thermodynamics, the topic has continued to fascinate engineers, physicists, chemists, computer scientists, mathematicians and philosophers. The topic of entropy is very much alive, as witnessed by the highly cited proceedings of...

We have produced extended series for two-dimensional prudent polygons, based on a transfer matrix algorithm of complexity O(n5), for a series of n-step polygons. For prudent polygons in two dimensions we find the growth constant to be smaller than that for the corresponding walks, and by considering three distinct subclasses of prudent walks and po...

This unique book gives a comprehensive account of new mathematical tools used to solve polygon problems. In the 20th and 21st centuries, many problems in mathematics, theoretical physics and theoretical chemistry and more recently in molecular biology and bio-informatics can be expressed as counting problems, in which specified graphs, or shapes, a...

The problems discussed in this book, particularly that of counting the number of polygons and polyominoes in two dimensions, either by perimeter or area, seems so simple to state that it seems surprising that they haven't been exactly solved. The counting problem is so simple in concept that it can be fully explained to any schoolchild, yet it seem...

Purpose This paper aims to provide a new quantitative methodology for predicting turning points and trends in financial markets time series based on information‐gap decision theory. Design/methodology/approach Uncertainty in future returns from financial markets is modeled using information‐gap decision theory. The robustness function, which measu...

We calculate very long low- and high-temperature series for the susceptibility $\chi$ of the square lattice Ising model as well as very long series for the five-particle contribution $\chi^{(5)}$ and six-particle contribution $\chi^{(6)}$. These calculations have been made possible by the use of highly optimized polynomial time modular algorithms a...

We have produced extended series for prudent self-avoiding walks on the square lattice. These are subsets of self-avoiding walks. We conjecture the exact growth constant and critical exponent for the walks, and show that the (anisotropic) generating function is almost certainly not differentiably-finite.

We calculate very long low- and high-temperature series for the susceptibility $\chi$ of the square lattice Ising model as well as very long series for the five-particle contribution $\chi^{(5)}$ and six-particle contribution $\chi^{(6)}$. These calculations have been made possible by the use of highly optimized polynomial time modular algorithms a...

Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their `concavity index', $m$. Such polygons are called \emph{$m$-convex} polygons and are characterised by having up to $m$ indentations in their perimeter. We first describe how we conjectured the (isotropic) generatin...

We study the conformations of polymer chains in a poor solvent, with and without bending rigidity, by means of a simple statistical mechanics model. This model can be exactly solved for chains of length up to N=55 using exact enumeration techniques. We analyze in details the differences between the constant force and constant distance ensembles for...

Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their `concavity index', $m$. Such polygons are called \emph{$m$-convex} polygons and are characterised by having up to $m$ indentations in the side. We use a `divide and conquer' approach, factorising 2-convex polygons...

A statistical mechanical description of flexible and semiflexible polymer chains in a poor solvent is developed in the constant force and constant distance ensembles. We predict the existence of many intermediate states at low temperatures stabilized by the force. A unified response to pulling and compressing forces has been obtained in the constan...

Punctured polygons are polygons with internal holes which are also polygons. The external and internal polygons are of the same type, and they are mutually as well as self-avoiding. Based on an assumption about the limiting area distribution for unpunctured polygons, we rigorously analyse the effect of a finite number of punctures on the limiting a...

Using a simple transfer matrix approach we have derived long series expansions for the perimeter generating functions of both three-choice polygons and punctured staircase polygons. In both cases we find that all the known terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 8. We report on an ana...

One partly solvable and two solvable models of polygons are discussed. Using a simple transfer matrix approach Iwan Jensen has derived very long series expansions for the perimeter generating function of both three-choice polygons and punctured staircase polygons. In both cases it is found that all the terms in the generating function can be reprod...

Using a simple transfer matrix approach we have derived very long series expansions for the perimeter generating function of punctured staircase polygons (staircase polygons with a single internal staircase hole). We find that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 8. We perf...

We study the correction-to-scaling exponents for the two-dimensional self-avoiding walk, using a combination of series-extrapolation and Monte Carlo methods. We enumerate all self-avoiding walks up to 59 steps on the square lattice, and up to 40 steps on the triangular lattice, measuring the mean-square end-to-end distance, the mean-square radius o...

Using a simple transfer matrix approach we have derived very long series expansions for the perimeter generating function of three-choice polygons. We find that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 8. We perform an analysis of the properties of the differential equation.

We study a restricted class of self-avoiding walks (SAW) which start at the origin (0, 0), end at $(L, L)$, and are entirely contained in the square $[0, L] \times [0, L]$ on the square lattice ${\mathbb Z}^2$. The number of distinct walks is known to grow as $\lambda^{L^2+o(L^2)}$. We estimate $\lambda = 1.744550 \pm 0.000005$ as well as obtaining...

We investigate the solvability of a variety of well-known problems in lattice statistical mechanics. We provide a new numerical procedure which enables one to conjecture whether the solution falls into a class of functions calleddifferentiably finite functions. Almost all solved problems fall into this class. The fact that one can conjecture whethe...

Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their ‘concavity index’, m. Such polygons are called m-convex polygons. We first use the inclusion–exclusion principle to rederive the known generating function for 1-convex self-avoiding polygons (SAPs). We then use ou...

We find the generating function of self-avoiding walks (SAWs) and trails on a semi-regular lattice called the 3.122 lattice in terms of the generating functions of simple graphs, such as SAWs, self-avoiding polygons and tadpole graphs on the hexagonal lattice. Since the growth constant for these graphs is known on the hexagonal lattice we can find...

We analyse new exact enumeration data for self-avoiding polygons, counted by perimeter and area on the square, triangular and hexagonal lattices. In extending earlier analyses, we focus on the perimeter moments in the vicinity of the bicritical point. We also consider the shape of the critical curve near the bicritical point, which describes the cr...

A combination of the refined finite lattice method and transfer matrices allows a radical increase in the computer enumeration of polyominoes on the hexagonal lattice (equivalently, site clusters on the triangular lattice), pn with n hexagons. We obtain pn for n⩽35. We prove that pn=τn+o(n), obtain the bounds 4.8049⩽τ⩽5.9047, and estimate that τ=5....

The finite lattice method of series expansions is used to calculate a number of extended series for susceptibilities of the three- and four-state Potts models on the square lattice. We analyse all these series and estimate their critical amplitudes. We resolve an uncertainty in amplitude ratios for high- to low-temperature susceptibilities, and thu...

Exactly solvable models of planar polygons, weighted by perimeter and area, have deepened our understanding of the critical behaviour of polygon models in recent years. Based on these results, we derive a conjecture for the exact form of the critical scaling function for planar self-avoiding polygons. The validity of this conjecture was recently te...

Poland-Scheraga models were introduced to describe the DNA denaturation transition. We give a rigorous and refined discussion of a family of these models. We derive possible scaling functions in the neighborhood of the phase transition point and review common examples. We introduce a self-avoiding Poland-Scheraga model displaying a first order phas...

We enumerate self-avoiding walks and polygons, counted by perimeter, on the quasiperiodic rhombic Penrose and Ammann-Beenker tilings, thereby considerably extending previous results. In contrast to similar problems on regular lattices, these numbers depend on the chosen start vertex. We compare different ways of counting and demonstrate that suitab...

For the study of Ising models of general spin S on the square lattice, we have combined our recently extended high-temperature expansions with the low-temperature expansions derived some time ago by Enting, Guttmann, and Jensen. We have computed various critical parameters and improved the estimates of others. Moreover, the properties of hyperscali...

For the study of Ising models of general spin S on the square lattice, we have combined our recently extended high-temperature expansions with the low-temperature expansions derived some time ago by Enting, Guttmann and Jensen. We have computed for the first time various critical parameters and improved the estimates of others. Moreover the propert...

Using a transfer matrix method, we present some results for directed lattice walkers in a horizontal strip of finite width. Some cases with two walkers in a small width are solved exactly, as are a couple of cases with vicious walkers in a small width; a conjecture is made for a case with three walkers. We also derive the general transfer matrix fo...

Low temperature Ising model series in the usual variable z = exp(−4J/kT) are analysed. The positions of the singularities in the disk|z| <or=|zc| and the values of the corresponding critical exponents are determined for the simple cubic, body-centred cubic and face-centred cubic lattices. We obtain the following estimates of the physical critical e...

A method is developed for determining the critical point and the critical exponent from terms in the series expansion of a function. For low temperature Ising model series the method provides an alternative technique to the method of Padé approximants. Application to some high and low temperature series for the Ising model on a simple cubic and a d...

A self-avoiding polygon (SAP) on a graph is an elementary cycle. Counting SAPs on the hypercubic lattice ℤd withd≥2, is a well-known unsolved problem, which is studied both for its combinatorial and probabilistic interest and its connections with statistical mechanics. Of course, polygons on ℤd are defined up to a translation, and the relevant stat...

We derive exact and asymptotic results for the number of star and watermelon configurations of vicious walkers confined to lie between two impenetrable walls, as well as for the analogous problem for $\infty$-friendly walkers. Our proofs make use of results from symmetric function theory and the theory of basic hypergeometric series.

Low temperature series expansions for the Ising model with spin S=1 and S=3/2 are derived for a number of two and three dimensional lattices. For each spin-lattice combination the critical point is estimated, and then the critical exponents beta and gamma ' with their corresponding critical amplitudes are also estimated. The principal conclusion is...

The limiting ring closure probability index alpha for the self avoiding random walk problem in two and three dimensions is investigated. It is concluded that this index is greater than the initial ring closure probability index, and that alpha >or=2 for both two and three dimensional lattices.

We introduce a model of friendly walkers which generalises the well-known vicious walker model. Friendly walkers refers to a model in which any number P of directed lattice paths, starting at adjacent lattice sites, simultaneously proceed in one of the allowed lattice directions. In the case of n-friendly walkers the paths may stay together for n v...

The perimeter and area generating functions of exactly solvable polygon models satisfy q-functional equations, where q is the area variable. The behaviour in the vicinity of the point where the perimeter generating function diverges can often be described by a scaling function. We develop the method of q-linear approximants in order to extract the...

We analyze new data for self-avoiding polygons, on the square and triangular lattices, enumerated by both perimeter and area, providing evidence that the scaling function is the logarithm of an Airy function. The results imply universal amplitude combinations for all area moments and suggest that rooted self-avoiding polygons may satisfy a $q$-alge...

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 wit...

We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N^6) to determine its first N coefficients. As a result, we have generated and analyzed...

We report computations of the short- and long-distance (scaling) contributions to the square-lattice Ising susceptibility. Both computations rely on summation of correlation functions, obtained using nonlinear partial difference equations. In terms of a temperature variable tau, linear in T/Tc-1, the short-distance terms have the form tau(p)(ln/tau...

New data for the number of selfavoiding walks and selfavoiding returns to the origin on two and three dimensional lattices are presented and studied numerically by the ratio method. Estimates for the critical attrition and critical indices are given. For a loose-packed lattice the selfavoiding walk generating function appears to have a singularity...

A new recurrence relation method for analysing the singular behaviour of series expansions is described. It is shown using plausible nonrigorous arguments that the method can resolve logarithmic singularities and finite cusp singularities.