Related papers: Random walk on a discrete torus and random interla…

Let $\xi(k,n)$ be the local time of a simple symmetric random walk on the line. We give a strong approximation of the centered local time process $\xi(k,n)-\xi(0,n)$ in terms of a Wiener sheet and an independent Wiener process, time changed…

In this paper, we consider fixed edgelength $n$-step random walks in $\mathbb{R}^d$. We give an explicit construction for the closest closed equilateral random walk to almost any open equilateral random walk based on the geometric median,…

A random walk problem with particles on discrete double infinite linear grids is discussed. The model is based on the work of Montroll and others. A probability connected with the problem is given in the form of integrals containing…

This paper is about the construction of displacement interpolations on a discrete metric graph. Our approach is based on the approximation of any optimal transport problem whose cost function is a distance on a discrete graph by a sequence…

We study existence of percolation in the hierarchical group of order $N$, which is an ultrametric space, and transience and recurrence of random walks on the percolation clusters. The connection probability on the hierarchical group for two…

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…

We obtain the joint distribution $P_N (X, K|Z)$ of the location $X$ of a one-dimensional symmetric next neighbor random walk on the integer lattice, and the number of times the walk has visited a specified site $Z$. This distribution has a…

We employ a simple and accurate dimensional interpolation formula for the shapes of random walks at $D=3$ and $D=2$ based on the analytically known solutions at both limits $D=\infty$ and $D=1$. The results obtained for the radii of…

We study the asymptotic behavior of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step…

We consider a nearest neighbor random walk on the one-dimensional integer lattice with drift towards the origin determined by an asymptotically vanishing function of the number of visits to zero. We show the existence of distinct regimes…

We prove for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n has the order of square root of n. Moment or symmetry assumptions are not necessary. In removing…

We consider a one-dimensional Brownian motion of fixed duration $T$. Using a path-integral technique, we compute exactly the probability distribution of the difference $\tau=t_{\min}-t_{\max}$ between the time $t_{\min}$ of the global…

Let $\rho$ be a probability measure on $\mathrm{SL}\_d(\mathbb{Z})$ and consider the random walk defined by $\rho$ on the torus $\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$. Bourgain, Furmann, Lindenstrauss and Mozes proved that under an…

Consider a continuous time random walk in $\mathbb{Z}$ with independent and exponentially distributed jumps $\pm1$. The model in this paper consists in an infinite number of such random walks starting from the complement of…

We consider a random walk on a multidimensional integer lattice with random bounds on local times, conditioned on the event that it hits a high level before its death. We introduce an auxiliary "core" process that has a regenerative…

We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by $N$) versions $D_N\subseteq \mathbb Z^2$ of bounded open domains $D\subseteq \mathbb R^2$. Upon exit from $D_N$, the walk lands on a…

In this article we obtain uniform estimates on the absorption of Brownian motion by porous interfaces surrounding a compact set. An important ingredient is the construction of certain resonance sets, which are hard to avoid for Brownian…

We consider the model of Branching Interlacements, introduced by Zhu, which is a natural analogue of Sznitman's Random Interlacements model, where the random walk trajectories are replaced by ranges of some suitable tree-indexed random…

We establish recurrence criteria for sums of independent random variables which take values in Euclidean lattices of varying dimension. In particular, we describe transient inhomogenous random walks in the plane which interlace two…

Consider a random walk on a tree $G=(V,E)$. For $v,w \in V$, let the hitting time $H(v,w)$ denote the expected number of steps required for the random walk started at $v$ to reach $w$, and let $\pi_v = \mathrm{deg}(v)/2|E|$ denote the…