Related papers: On the probability that self-avoiding walk ends at…

For d at least two and integer n, let c_n = c_n(d) denote the number of length n self-avoiding walks beginning at the origin in the integer lattice Z^d, and, for even n, let p_n = p_n(d) denote the number of length n self-avoiding polygons…

For $d \geq 2$ and $n \in \mathbb{N}$, let $\mathsf{W}_n$ denote the uniform law on self-avoiding walks of length $n$ beginning at the origin in the nearest-neighbour integer lattice $\mathbb{Z}^d$, and write $\Gamma$ for a…

For $d \geq 2$ and $n \in \mathbb{N}$, let $\mathsf{W}_n$ denote the uniform law on self-avoiding walks beginning at the origin in the integer lattice $\mathbb{Z}^d$, and write $\Gamma$ for a $\mathsf{W}_n$-distributed walk. We show that…

We prove that on any transitive graph $G$ with infinitely many ends, a self-avoiding walk of length $n$ is ballistic with extremely high probability, in the sense that there exist constants $c,t>0$ such that $\mathbb{P}_n(d_G(w_0,w_n)\geq…

We consider simple random walk on Z^d, d bigger or equal to 3. Motivated by the work of A.-S. Sznitman and the author in arXiv:1304.7477 and arXiv:1310.2177, we investigate the asymptotic behaviour of the probability that a large body gets…

We prove that self-avoiding walk on Z^d is sub-ballistic in any dimension d at least two. That is, writing ||u|| for the Euclidean norm of u \in Z^d, and SAW_n for the uniform measure on self-avoiding walks gamma:{0,...,n} \to Z^d for which…

In this paper we find an upper bound for the probability that a $3$ dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball of radius $N$. For $d\ge 4$, it has been shown in…

We prove quantitative sub-ballisticity for the self-avoiding walk on the hexagonal lattice. Namely, we show that with high probability a self-avoiding walk of length $n$ does not exit a ball of radius $O(n/\log{n})$. Previously, only a…

We study models of continuous time, symmetric, $\Z^d$-valued random walks in random environments. One of our aims is to derive estimates on the decay of transition probabilities in a case where a uniform ellipticity assumption is absent. We…

Let $(Z_n)_{n\in\N}$ be a $d$-dimensional {\it random walk in random scenery}, i.e., $Z_n=\sum_{k=0}^{n-1}Y(S_k)$ with $(S_k)_{k\in\N_0}$ a random walk in $\Z^d$ and $(Y(z))_{z\in\Z^d}$ an i.i.d. scenery, independent of the walk. The…

In this paper we study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball. We show that among all such paths,…

We consider the biased random walk on a tree constructed from the set of finite self-avoiding walks on a lattice, and use it to construct probability measures on infinite self-avoiding walks. The limit measure (if it exists) obtained when…

Suppose that $X$ is a simple random walk on $\Z_n^d$ for $d \geq 3$ and, for each $t$, we let $\U(t)$ consist of those $x \in \Z_n^d$ which have not been visited by $X$ by time $t$. Let $\tcov$ be the expected amount of time that it takes…

We consider random self-avoiding walks between two points on the boundary of a finite subdomain of Z^d (the probability of a self-avoiding trajectory gamma is proportional to mu^{-length(gamma)}). We show that the random trajectory becomes…

Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$, and run a simple random walk until hitting one of the obstacles. For $d\geq 2$ and $p$ strictly above the critical threshold for site percolation, we…

A simple symmetric random walk in the space $\mathbb{Z}^2$ is considered. The asymptotic behavior as the number of jumps tends to infinity of the probability that a fixed edge of the random walk lies in the polygon that forms the boundary…

The kinetics of annihilating random walks in one dimension, with the half-line x>0 initially filled, is investigated. The survival probability of the nth particle from the interface exhibits power-law decay, S_n(t)~t^{-alpha_n}, with…

The decay of directional correlations in self-avoiding random walks on the square lattice is investigated. Analysis of exact enumerations and Monte Carlo data suggest that the correlation between the directions of the first step and the…

We consider Sinai's random walk in random environment $(S_n)_{n\in\mathbb{N}}$. We prove a local limit theorem for $(S_n)_{n\in\mathbb{N}}$ under the annealed law $\mathbb{P}$. As a consequence, we get an equivalent for the annealed…

We give sharp, uniform estimates for the probability that a random walk of n steps on the reals avoids a half-line [y,infinity) given that it ends at the point x. The estimates hold for general continuous or lattice distributions provided…