Related papers: Exponents of interchain correlation for self-avoid…

The phase diagram and surface critical behaviour of the vertex-interacting self-avoiding walk are examined using transfer matrix methods extended using DMRG and coupled with finite-size scaling. Particular attention is paid to the critical…

The surface critical behaviour of the interacting self-avoiding trail is examined using transfer matrix methods coupled with finite-size scaling. Particular attention is paid to the critical exponents at the ordinary and special points…

These lecture notes provide a rapid introduction to a number of rigorous results on self-avoiding walks, with emphasis on the critical behaviour. Following an introductory overview of the central problems, an account is given of the…

We calculate the connective constant for self-avoiding walks on the simple cubic lattice to unprecedented accuracy, using a novel application of the pivot algorithm. We estimate that \mu = 4.684 039 931(27). Our method also provides…

We define a new ensemble for self-avoiding walks in the upper half-plane, the fixed irredicible bridge ensemble, by considering self-avoiding walks in the upper half-plane up to their $n$-th bridge height, $Y_n$, and scaling the walk by…

This article is dedicated to the study of the 2-dimensional interacting prudent self-avoiding walk (referred to by the acronym IPSAW) and in particular to its collapse transition. The interaction intensity is denoted by $\beta>0$ and the…

For many stochastic processes, the probability $S(t)$ of not-having reached a target in unbounded space up to time $t$ follows a slow algebraic decay at long times, $S(t)\sim S_0/t^\theta$. This is typically the case of symmetric compact…

We consider the equiprobable distribution of spanning trees on the square lattice. All bonds of each tree can be oriented uniquely with respect to an arbitrary chosen site called the root. The problem of predecessors is finding the…

We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in dimensions $d>4$, in the vicinity of the critical point, and prove an upper bound…

Non-local twist operators are introduced for the O(n) and Q-state Potts models in two dimensions which, in the limits n -> 0 (resp. Q -> 1) count the numbers of self-avoiding loops (resp. percolation clusters) surrounding a given point.…

We prove that the set of thick points of the log-correlated Gaussian field contains an unbounded path in sufficiently high dimensions. This contrasts with the two-dimensional case, where Aru, Papon, and Powell (2023) showed that the set of…

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…

Counting the number of N-step self-avoiding walks (SAWs) on a lattice is one of the most difficult problems of enumerative combinatorics. Once we give up calculating the exact number of them, however, we have a chance to apply powerful…

We study the self-avoiding walk on complex fractal networks called the (u,v)-flower by mapping it to the N-vector model in a generating function formalism. First, we analytically calculate the critical exponent {\nu} and the connective…

For independent nearest-neighbour bond percolation on Z^d with d >> 6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling…

We reduce the problem of counting self-avoiding walks in the square lattice to a problem of counting the number of integral points in multidimensional domains. We obtain an asymptotic estimate of the number of self-avoiding walks of length…

This is the first of two papers on the critical behaviour of bond percolation models in high dimensions. In this paper, we obtain strong joint control of the critical exponents eta and delta, for the nearest-neighbour model in very high…

We consider a $d$-dimensional correlated percolation problem of sites {\em not} visited by a random walk on a hypercubic lattice $L^d$ for $d=3$, 4 and 5. The length of the random walk is ${\cal N}=uL^d$. Close to the critical value…

We define multiple chordal SLEs in a simply connected domain by considering a natural configurational measure on paths. We show how to construct these measures so that they are conformally covariant and satisfy certain boundary perturbation…

We investigate a model of continuous-time simple random walk paths in $\mathbb{Z}^d$ undergoing two competing interactions: an attractive one towards the large values of a random potential, and a self-repellent one in the spirit of the…