Related papers: Poisson-Dirichlet branching random walks

We consider branching random walks on the Euclidean lattice in dimensions five and higher. In this non-Markovian setting, we first obtain a relationship between the equilibrium measure and Green's function, in the form of an approximate…

Let $\{Z_n\}_{n\geq 0 }$ be a critical $d$-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesgue measure on $\mathbb{R}^d$. Denote by…

We consider reflecting random walks on the nonnegative integers with drift of order 1/x at height x. We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of 0…

Consider the following computational problem: given a regular digraph $G=(V,E)$, two vertices $u,v \in V$, and a walk length $t\in \mathbb{N}$, estimate the probability that a random walk of length $t$ from $u$ ends at $v$ to within $\pm…

In this paper we consider a random walk in random environment on a tree and focus on the boundary case for the underlying branching potential. We study the range $R\_n$ of this walk up to time $n$ and obtain its correct asymptotic in…

We construct the conditional version of $k$ independent and identically distributed random walks on $\R$ given that they stay in strict order at all times. This is a generalisation of so-called non-colliding or non-intersecting random…

Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen at random with probability proportional to its weight. In the case where the total…

We consider the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p by studying three classes of…

We consider a quantum particle (walker) on a line who coherently chooses to jump to the left or right depending on the result of toss of a quantum coin. The lengths of the jumps are considered to be independent and identically distributed…

In this paper, we study the critical branching random walk in the critical dimension, $Z^4$. We provide the asymptotics of the probability of visiting a fixed finite subset and the range of the critical branching random walk conditioned on…

We consider discrete-time branching random walks with a radially symmetric distribution. Independently of each other individuals generate offspring whose relative locations are given by a copy of a radially symmetric point process…

We consider the P\'olya random walk in $\mathbb{Z}^2$. The paper establishes a number of results for the distributions and expectations of the number of usual (undirected) and specifically defined in the paper up- and down-directed…

We study random walks evolving in continuous time on a one-dimensional lattice where each site $x$ hosts a quenched random potential $U_x$. The potentials on different sites are independent, identically distributed Gaussian random…

We are interested in the randomly biased random walk on the supercritical Galton--Watson tree. Our attention is focused on a slow regime when the biased random walk $(X_n)$ is null recurrent, making a maximal displacement of order of…

We consider a two-speed branching random walk, which consists of two macroscopic stages with different reproduction laws. We prove that the centered maximum converges in law to a Gumbel variable with a random shift and the extremal process…

The biased random walk on supercritical Galton--Watson trees is known to exhibit a multiscale phenomenon in the slow regime: the maximal displacement of the walk in the first $n$ steps is of order $(\log n)^3$, whereas the typical…

For a simple (unbiased) random walk on a connected graph with $n$ vertices, the cover time (the expected number of steps it takes to visit all vertices) is at most $O(n^3)$. We consider locally biased random walks, in which the probability…

We investigate the first passage statistics of active continuous time random walks with Poissonian waiting time distribution on a one dimensional infinite lattice and a two dimensional infinite square lattice. We study the small and large…

In this expository note, we give a short derivation of the expected number of collisions between two independent simple random walkers on integer lattices. Adapting a Poissonization technique introduced by Lange, we express the collision…

We report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree. The term \textit{multiplexed} means that the model can be viewed as a nearest…