Related papers: Divisible sandpiles via random walks in random sce…

We study Abelian sandpiles on graphs of the form $G \times I$, where $G$ is an arbitrary finite connected graph, and $I \subset \Z$ is a finite interval. We show that for any fixed $G$ with at least two vertices, the stationary measures…

We show that Activated Random Walk on $\mathbb{Z}$ is explosive above criticality. That is, activating a single particle in a supercritical state of sleeping particles triggers an infinite avalanche of activity with positive probability.…

We consider random walk in Dirichlet random environment in ${\mathbf{Z}^d, d\ge 3}$, which corresponds to the case where the environment is constructed from i.i.d. transition probabilities at each vertex with a Dirichlet distribution with…

In this article we study collisions of two independent random walks on comb graphs $\mathrm{Comb}(\tilde{G},f)$ for a large class of recurrent planar graphs $\tilde{G}$ and profile functions $f$, the latter governing the length of vertical…

We prove that the wired uniform spanning forest exhibits mean-field behaviour on a very large class of graphs, including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph. Several of our…

We study driven-dissipative activated random walk with sleep probability $p$ on an $n$-vertex complete graph with a sink that traps jumping particles with probability $q_n$. We show that the number of sleeping particles $S_n$ left by the…

We prove that in any recurrent reversible random rooted graph, two independent simple random walks started at the same vertex collide infinitely often almost surely. This applies to the Uniform Infinite Planar Triangulation and…

We consider the stochastic sandpile model with uniform toppling rule on the integer line. During a uniform toppling, with probability $1/3$ one particle is sent to the right of the toppled vertex, with probability $1/3$ one particle is sent…

We prove that for a random walk on the real line whose increments have zero mean and are either integer-valued or spread out (i.e. the distributions of the steps of the walk are eventually non-singular), the Markov chain of overshoots above…

Let $G$ be a Cayley graph of a nonamenable group with spectral radius $\rho < 1$. It is known that branching random walk on $G$ with offspring distribution $\mu$ is transient, i.e., visits the origin at most finitely often almost surely, if…

We prove non-universality results for first-passage percolation on the configuration model with i.i.d. degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of…

We examine the stationary distribution of random walks on directed graphs. In particular, we focus on the {\em principal ratio}, which is the ratio of maximum to minimum values of vertices in the stationary distribution. We give an upper…

We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail behaviour with exponent $\alpha+1$, where $\alpha \in (1,2)$. The limiting components are…

We study a restricted-height version of the one-dimensional Oslo sandpile with conserved density, using periodic boundary conditions. Each site has a limiting height which can be either two or three. When a site reaches its limiting height…

We show that anomalous diffusion can result when the steps of a random walk are not statistically independent. We present an algorithm that counts all the possible paths of particles diffusing on random graphs with arbitrary degree…

Graph-limit theory focuses on the convergence of sequences of graphs when the number of nodes becomes arbitrarily large. This framework defines a continuous version of graphs allowing for the study of dynamical systems on very large graphs,…

In this paper we study $\infty$-Laplacian type diffusion equations in weighted graphs obtained as limit as $p\to \infty$ to two types of $p$-Laplacian evolution equations in such graphs. We propose these diffusion equations, that are…

Let $X$ be a graph with adjacency matrix $A$. The \textsl{continuous quantum walk} on $X$ is determined by the unitary matrices $U(t)=\exp(itA)$. If $X$ is the complete graph $K_n$ and $a\in V(X)$, then \[1-|U(t)_{a,a}|\le2/n. \] In a…

We study diffusion of particles in large-scale simulations of one-dimensional stochastic sandpiles, in both the restricted and unrestricted versions. The results indicate that the diffusion constant scales in the same manner as the activity…

We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are…