Related papers: Giant Component and Vacant Set for Random Walk on …
This work continues the investigation, initiated in a recent work by Benjamini and Sznitman, of percolative properties of the set of points not visited by a random walk on the discrete torus (Z/NZ)^d up to time uN^d in high dimension d. If…
We consider a simple random walk on a discrete torus (Z/NZ)^d with dimension d at least 3 and large side length N. For a fixed constant u > 0, we study the percolative properties of the vacant set, consisting of the set of vertices not…
We study the trajectory of a simple random walk on a d-regular graph with d>2 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these…
We consider the set of points visited by the random walk on the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, for $d \geq 3$, at times of order $uN^d$, for a parameter $u>0$ in the large-$N$ limit. We prove that the vacant set left by the…
We study the simple random walk on the giant component of a supercritical Erd\H{o}s-R\'enyi random graph on $n$ vertices, in particular the so-called vacant set at level $u$, the complement of the trajectory of the random walk run up to a…
We consider the simple random walk on a random d-regular graph with n vertices, and investigate percolative properties of the set of vertices not visited by the walk until time un, where u > 0 is a fixed positive parameter. It was shown in…
We consider random walks on several classes of graphs and explore the likely structure of the vacant set, i.e. the set of unvisited vertices. Let \Gamma(t) be the subgraph induced by the vacant set of the walk at step t. We show that for…
We consider one or more independent random walks on the $d\ge 3$ dimensional discrete torus. The walks start from vertices chosen independently and uniformly at random. We analyze the fluctuation behavior of the size of some random sets…
The model of random interlacements is a one-parameter family $\mathcal I^u,$ $u \ge 0,$ of random subsets of $\mathbb{Z}^d,$ which locally describes the trace of simple random walk on a $d$-dimensional torus run up to time $u$ times its…
We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter the picture. This model describes in the…
We explore some of the connections between the local picture left by the trace of simple random walk on a discrete cylinder with base a d-dimensional torus, d at least 2, of side-length N running for times of order N^{2d} and the model of…
Random interlacements at level u is a one parameter family of connected random subsets of Z^d, d>=3 introduced in arXiv:0704.2560. Its complement, the vacant set at level u, exhibits a non-trivial percolation phase transition in u, as shown…
We consider a random walk on a $d$-regular graph $G$ where $d\to\infty$ and $G$ satisfies certain conditions. Our prime example is the $d$-dimensional hypercube, which has $n=2^d$ vertices. We explore the likely component structure of the…
We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus (Z/NZ)^d, d >= 3, until u N^d time steps, u > 0, and the model of random interlacements recently introduced by Sznitman. In…
We consider the percolation problem of sites on an $L\times L$ square lattice with periodic boundary conditions which were unvisited by a random walk of $N=uL^2$ steps, i.e. are vacant. Most of the results are obtained from numerical…
How does removal of sites by a random walk lead to blockage of percolation? To study this problem of correlated site percolation, we consider a random walk (RW) of $N=uL^d$ steps on a $d$-dimensional hypercubic lattice of size $L^d$ (with…
In this paper we establish some properties of percolation for the vacant set of random interlacements, for d at least 5 and small intensity u. The model of random interlacements was first introduced by A.S. Sznitman in arXiv:0704.2560. It…
We consider simple random walk on a discrete cylinder with base a large d-dimensional torus of side-length N, when d is two or more. We develop a stochastic domination control on the local picture left by the random walk in boxes of…
For $d\ge 3$ we construct a new coupling of the trace left by a random walk on a large $d$-dimensional discrete torus with the random interlacements on $\mathbb Z^d$. This coupling has the advantage of working up to macroscopic subsets of…
Given a discrete random walk on a finite graph $G$, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step $t$.%These sets induce subgraphs of the underlying graph.…