Related papers: Diffusive bounds for the critical density of activ…

We study critical branching random walks (BRWs) $U^{(n)}$ on~$\mathbb{Z}_{+}$ where for each $n$, the displacement of an offspring from its parent has drift~$2\beta/\sqrt{n}$ towards the origin and reflection at the origin. We prove that…

We prove that for the Activated Random Walks model on transitive unimodular graphs, if there is fixation, then every particle eventually fixates, almost surely. We deduce that the critical density is at most 1. Our methods apply for much…

We study the critical branching random walk on $\mathbb{Z}^d$ started from a distant point $x$ and conditioned to hit some compact set $K$ in $\mathbb{Z}^d$. We are interested in the occupation time in $K$ and present its asymptotic…

We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of $\lfloor sN\rfloor$ branching random walks, viewed as a function-valued,…

The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter $\lambda$. There is a threshold for $\lambda$, which is called $\lambda_w$, that separates almost sure global extinction from global…

We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time $n$. Assuming that the moment of order $2+\delta$ is…

We consider critical site percolation ($p=p_c=1/2$) on the triangular lattice $\mathbf{T}$ in two dimensions. We show that the simple random walk on the clusters of open vertices converges in the scaling limit to a continuous diffusion…

Let $\{X_n\}_{n\in\mathbb{N}}$ be a sequence of i.i.d. random variables in $\mathbb{Z}^d$. Let $S_k=X_1+...+X_k$ and $Y_n(t)$ be the continuous process on $[0,1]$ for which $Y_n(k/n)=S_k/\sqrt{n}$ $k=1,...,n$ and which is linearly…

In the present paper, as a continuation of our preceding paper [10], we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a viewpoint of discrete geometric analysis…

We consider a random walk $(Z^{(1)}_n, ..., Z^{(K+1)}_n) \in \mathbb{Z}^{K+1}$ with the constraint that each coordinate of the walk is at distance one from the following one. In this paper, we show that this random walk is slowed down by a…

In this paper, we study the dynamics of a random walker diffusing on a disordered one-dimensional lattice with random trappings. The distribution of escape probabilities is computed exactly for any strength of the disorder. These…

We consider a multidimensional random walk in a product random environment with bounded steps, transience in some spatial direction, and high enough moments on the regeneration time. We prove an invariance principle, or functional central…

We study the range $R_n$ of a random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ indexed by a random tree with $n$ vertices. Under the assumption that the random walk is centered and has finite fourth moments, we prove in dimension…

We derive a functional central limit theorem for the excursion of a random walk conditioned on sweeping a prescribed geometric area. We assume that the increments of the random walk are integer-valued, centered, with a third moment equal to…

Percolation theory has become a useful tool for the analysis of large-scale wireless networks. We investigate the fundamental problem of characterizing the critical density $\lambda_c^{(d)}$ for $d$-dimensional Poisson random geometric…

Let $\{Z_{m},m\geq 0\}$ be a critical branching process in random environment and $\{S_{m},m\geq 0\}$ be its associated random walk. Assuming that the increments distribution of the associated random walk belongs without centering to the…

Starting with a percolation model in $\Z^d$ in the subcritical regime, we consider a random walk described as follows: the probability of transition from $x$ to $y$ is proportional to some function $f$ of the size of the cluster of $y$.…

We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is…

We establish a novel type of connection between random walks and analytic number theory. Working with a random walk on the circle group $\mathbb{R}/\mathbb{Z}$ in which each step is a random integer multiple of a given quadratic irrational…

We show that for a weakly dense subset of the domain of attraction of a positive stable random variable of index $0<\alpha<1$($DOA\left(\alpha\right))$ the functional stable convergence is a time-changed renewal convergence of distribution…