Related papers: Cover Times for Brownian Motion and Random Walks i…

We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line $ (- \infty,0] \times {0}$ before time $n$. Let $X^{(1)}=(X_{1},X_{2})$ be the increment of the two-dimensional random…

We consider dynamical percolation on the $d$-dimensional discrete torus of side length $n$, $\mathbb{Z}_n^d$, where each edge refreshes its status at rate $\mu=\mu_n\le 1/2$ to be open with probability $p$. We study random walk on the…

In a flat 2-torus with a disk of diameter $r$ removed, let $\Phi_r(t)$ be the distribution of free-path lengths (the probability that a segment of length larger than $t$ with uniformly distributed origin and direction does not meet the…

In the simplest case, consider a $\mathbb{Z}^d$-periodic ($d \geq 3$) arrangement of balls of radii < 1/2, and select a random direction and point (outside the balls). According to Dettmann's first conjecture, the probability that the so…

A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence,…

We develop techniques to obtain rigorous bounds on the behaviour of random walks on combs. Using these bounds we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but…

We study the problem of searching for a fixed path $\epsilon_0\epsilon_1\cdots\epsilon_l$ on a network through random walks. We analyze the first hitting time of tracking the path, and obtain exact expression of mean first hitting time…

Convergence of directed forests, spanning on random subsets of lattices or on point processes, towards the Brownian web has made the subject of an abundant literature, a large part of which relies on a criterion proposed by Fontes, Isopi,…

We consider an obliquely reflected Brownian motion $Z$ with positive drift in a quadrant stopped at time $T$, where $T:=\inf \{ t>0 : Z(t)=(0,0) \}$ is the first hitting time of the origin. Such a process can be defined even in the…

In this paper we study a random walk on an affine building of type $\tilde{A}_r$, whose radial part, when suitably normalized, converges to the Brownian motion of the Weyl chamber. This gives a new discrete approximation of this process,…

We consider a model for random walks on random environments (RWRE) with random subset of the d-dimensional Euclidean lattice as the vertices, and uniform transition probabilities on 2d points (two "coordinate nearest points" in each of the…

In the present paper, we study an explicit effect of non-symmetry on asymptotics of the $n$-step transition probability as $n\rightarrow \infty$ for a class of non-symmetric random walks on the triangular lattice. Realizing the triangular…

We show that the critical density of the Activated Random Walk model on $\mathbb{Z}^d$ is strictly less than one when the sleep rate $\lambda$ is small enough, and tends to $0$ when $\lambda\to 0$, in any dimension $d\geqslant 1$. As far as…

We derive a three-term asymptotic expansion for the expected lifetime of Brownian motion and for the torsional rigidity on thin domains in R^n, and a two-term expansion for the maximum (and corresponding maximizer) of the expected lifetime.…

We consider a discrete-time random motion, Markov chain on the Poincar\'{e} disk. In the basic variant of the model a particle moves along certain circular arcs within the disk, its location is determined by a composition of random…

We study a discrete-time random walk on the non-negative integers, such that when 0 is reached a jump occurs to an arbitrary location, with given probabilities. We obtain an asymptotic formula for the expected position at large times, in…

The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. We consider two independent random walks on a hyper-cubic lattice and study ordering probabilities associated with these…

We consider random paths on a square lattice which take a left or a right turn at every vertex. The possible turns are taken with equal probability, except at a vertex which has been visited before. In such case the vertex is left via the…

Brownian motion and scaled and interpolated simple random walk can be jointly embedded in a probability space in such a way that almost surely the $n$-step walk is within a uniform distance $O(n^{-1/2}\log n)$ of the Brownian path for all…

We study a space-time Brownian motion with drift B(t)=(t_0+t,y_0+W(t)+t) killed at the moving boundary of the cone {(t,x):0<x<t}. This article determines the parabolic Martin boundary and all harmonic functions associated with this process.…