Related papers: Almost sure convergence for stochastically biased …

The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this…

We prove CLTs for biased randomly trapped random walks in one dimension. In particular, we will establish an annealed invariance principal by considering a sequence of regeneration times under the assumption that the trapping times have…

We consider a one dimensional random walk in random environment that is uniformly biased to one direction. In addition to the transition probability, the jump rate of the random walk is assumed to be spatially inhomogeneous and random. We…

We consider a branching random walk on a $d$-ary tree of height $n$ ($n \in \mathbb{N}$), under the presence of a hard wall which restricts each value to be positive, where $d$ is a natural number satisfying $d\geqslant2$. The question of…

We prove a law of large numbers for the range of rotor walks with random initial configuration on regular trees and on Galton-Watson trees. More precisely, we show that on the classes of trees under consideration, even in the case when the…

In this article, we study the maximal displacement of critical branching random walk in random environment. Let $M_n$ be the maximal displacement of a particle in generation $n$, and $Z_n$ be the total population in generation $n$, $M$ be…

We study spanning trees on Sierpinski graphs (i.e., finite approximations to the Sierpinski gasket) that are chosen uniformly at random. We construct a joint probability space for uniform spanning trees on every finite Sierpinski graph and…

Let $b$ be an integer greater than 1 and let $W^{\ee}=(W^{\ee}_n; n\geq 0)$ be a random walk on the $b$-ary rooted tree $\U_b$, starting at the root, going up (resp. down) with probability $1/2+\epsilon$ (resp. $1/2 -\epsilon$), $\epsilon…

In this article, we study the maximal displacement in a branching random walk. We prove that its asymptotic behaviour consists in a first almost sure ballistic term, a negative logarithmic correction in probability and stochastically…

Let $\{S_n\}$ be a random walk in the domain of attraction of a stable law $\mathcal{Y}$, i.e. there exists a sequence of positive real numbers $(a_n)$ such that $S_n/a_n$ converges in law to $\mathcal{Y}$. Our main result is that the…

We study a branching random walk with independent and identically distributed, heavy tailed displacements. The offspring law is supercritical and satisfies the Kesten-Stigum condition. We treat the case when the law of the displacements…

Let $(W_n(\theta))_{n \in \mathbb{N}_0}$ be Biggins' martingale associated with a supercritical branching random walk, and let $W(\theta)$ be its almost sure limit. Under a natural condition for the offspring point process in the branching…

We consider biased random walks in a one-dimensional percolation model. This model goes back to Axelson-Fisk and H\"aggstr\"om and exhibits the same phase transition as biased random walk on the infinite cluster of supercritical Bernoulli…

Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope $\gamma-\epsilon$, where $\gamma$…

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 discuss various forms of convergence of the vicinity of a uniformly at random selected vertex in random simply generated trees, as the size tends to infinity. For the standard case of a critical Galton-Watson tree conditioned to be large…

We study the maximal displacement of branching random walks in a class of time inhomogeneous environments. Specifically, binary branching random walks with Gaussian increments will be considered, where the variances of the increments change…

We consider a Branching Random Walk on $\R$ whose step size decreases by a fixed factor, $0<b<1$, with each turn. This process generates a random probability measure on $\R$, that is, the limit of uniform distribution among the $2^n$…

We study the random planar maps obtained from supercritical Galton--Watson trees by adding the horizontal connections between successive vertices at each level. These are the hyperbolic analog of the maps studied by Curien, Hutchcroft and…

Given a connected graph $G$ with some subset of its vertices excited and a fixed target vertex, in the geodesic-biased random walk on $G$, a random walker moves as follows: from an unexcited vertex, she moves to a uniformly random…