Related papers: Branching capacity of a random walk range
We introduce a Gibbs measure on nearest-neighbour paths of length $t$ in the Euclidean $d$-dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature $\beta$. We prove…
We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant…
We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green's function is the probability that the LERW passes through a given edge in the…
We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in ${\Bbb R}^d$ or ${\Bbb Z}^d$. The first class consists of random walks on ${\Bbb Z}^d$ in divergence-free random drift field,…
We consider a recurrent random walk of i.i.d. increments on the one-dimensional integer lattice and obtain a formula relating the hitting distribution of a half-line with the potential function, $a(x)$, of the random walk. Applying it, we…
We study limit laws for simple random walks on supercritical long range percolation clusters on $\Z^d, d \geq 1$. For the long range percolation model, the probability that two vertices $x, y$ are connected behaves asymptotically as…
We consider simple exclusion processes on Z for which the underlying random walk has a finite first moment and a non-zero mean and whose initial distributions are product measures with different densities to the left and to the right of the…
The Airy line ensemble is a random collection of continuous ordered paths that plays an important role within random matrix theory and the Kardar-Parisi-Zhang universality class. The aim of this paper is to prove a universality property of…
We consider a random walk on the Manhattan lattice. The walker must follow the orientations of the bonds in this lattice, and the walker is not allowed to visit a site more than once. When both possible steps are allowed, the walker chooses…
A directed polymer is allowed to branch, with configurations determined by global energy optimization and disorder. A finite size scaling analysis in 2D shows that, if disorder makes branching more and more favorable, a critical transition…
We study limit laws for simple random walks on supercritical long-range percolation clusters on the integer lattice. For the long range percolation model, the probability that two vertices are connected behaves asymptotically as a negative…
We prove that a law of large numbers and a central limit theorem hold for the excited random walk model in every dimension $d \geq 2$.
We consider random walks in a random environment that is given by i.i.d. Dirichlet distributions at each vertex of Z^d or, equivalently, oriented edge reinforced random walks on Z^d. The parameters of the distribution are a 2d-uplet of…
One can define a random walk on a hypercubic lattice in a space of integer dimension $D$. For such a process formulas can be derived that express the probability of certain events, such as the chance of returning to the origin after a given…
In this article we study a one dimensional model for a polymer in a poor solvent: the random walk on $\mathbb{Z}$ penalized by its range. More precisely, we consider a Gibbs transformation of the law of the simple symmmetric random walk by…
We outline basic properties of a symmetric random walk in one dimension, in which the length of the nth step equals lambda^n, with lambda<1. As the number of steps N-->oo, the probability that the endpoint is at x, P_{lambda}(x;N),…
In this paper, we deal with the inner boundary of random walk range, that is, the set of those points in a random walk range which have at least one neighbor site outside the range. If $L_n$ be the number of the inner boundary points of…
For a one-dimensional random walk in random scenery (RWRS) on Z, we determine its quenched weak limits by applying Strassen's functional law of the iterated logarithm. As a consequence, conditioned on the random scenery, the one-dimensional…
We consider a ballistic random walk in an i.i.d. random environment that does not allow retreating in a certain fixed direction. We prove an invariance principle (functional central limit theorem) under almost every fixed environment. The…
We consider the branching random walk in random environment with a random absorption wall. When we add this barrier, we discuss some topics related to the survival probability. We assume that the random environment is i.i.d., $S_i$ is a…