Related papers: Universal Amplitude Combinations for Self-Avoiding…
The statistics of self-avoiding random walks have been used to model polymer physics for decades. A self-avoiding walk that grows one step at a time on a lattice will eventually trap itself, which occurs after an average of 71 steps on a…
We calculate improved lower bounds for the connective constants for self-avoiding walks on the square, hexagonal, triangular, $(4.8^2)$, and $(3.12^2)$ lattices. The bound is found by Kesten's method of irreducible bridges. This involves…
We study the winding angles of random and self-avoiding walks on square and cubic lattices with number of steps $N$ ranging up to $10^7$. We show that the mean square winding angle $\langle\theta^2\rangle$ of random walks converges to the…
The statistics of a long closed self-avoiding walk (SAW) or polymer ring on a $ d $-dimensional lattice obeys hyperscaling. The combination $ p_N \left\langle R^2 \right\rangle^{ d/2}_N\mu^{ -N}, $ (where $ p_N $ is the number of…
We prove several rigorous results about the asymptotic behaviour of the numbers of polygons and self-avoiding walks confined to a square on the square lattice. Specifically we prove that the dominant asymptotic behaviour of polygons…
We introduce a self-avoiding walk model for which end-effects are completely eliminated. We enumerate the number of these walks for various lattices in dimensions two and three, and use these enumerations to study the properties of this…
We investigate active lattice walks: biased continuous time random walks which perform orientational diffusion between lattice directions in one and two spatial dimensions. We study the occupation probability of an arbitrary site on the…
A comprehensive numerical study of self-avoiding walks (SAW's) on randomly diluted lattices in two and three dimensions is carried out. The critical exponents $\nu$ and $\chi$ are calculated for various different occupation probabilities,…
Walks in a directed graph can be given a partially ordered structure that extends to possibly unconnected objects, called hikes. Studying the incidence algebra on this poset reveals unsuspected relations between walks and self-avoiding…
The number of free sites next to the end of a self-avoiding walk is known as the atmosphere. The average atmosphere can be related to the number of configurations. Here we study the distribution of atmospheres as a function of length and…
We experimentally demonstrate that the statistical properties of distances between pedestrians which are hindered from avoiding each other are described by the Gaussian Unitary Ensemble of random matrices. The same result has recently been…
We have derived analytic expressions for the amplitudes of correction terms in the critical free energy and inverse correlation length expansions for square, honeycomb, and plane-triangular lattices in strip geometry. We have found that…
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…
We study an homogeneous irreducible markovian random walk in a square lattice of arbitrary dimension, with an antisymmetric perturbation acting only in one point. We compute exactly spatial correction to the diffusive behaviour in the…
We describe a new algebraic technique, utilising transfer matrices, for enumerating self-avoiding lattice trails on the square lattice. We have enumerated trails to 31 steps, and find increased evidence that trails are in the self-avoiding…
We study two simple modifications of self-avoiding polygons. Osculating polygons are a super-set in which we allow the perimeter of the polygon to touch at a vertex. Neighbour-avoiding polygons are only allowed to have nearest neighbour…
For more than a century lattice random walks have been employed ubiquitously, both as a theoretical laboratory to develop intuition about more complex stochastic processes and as a tool to interpret a vast array of empirical observations.…
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…
We derive a series of results on random walks on a d-dimensional hypercubic lattice (lattice paths). We introduce the notions of terse and simple paths corresponding to the path having no backtracking parts (spikes). These paths label…
We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time…