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

A popular theory of self-organized criticality relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the abelian sandpile model…

We present a method which provides a unified framework for most stability theorems that have been proved in graph and hypergraph theory. Our main result reduces stability for a large class of hypergraph problems to the simpler question of…

Let $F\{dx\}$ be a relatively stable probability distribution on the whole real line and $S_n$ the random walk started at the origin with step distribution $F$. We obtain an exact asymptotic form of the Green measure $U\{x+dy\}=…

We study two random processes on an $n$-vertex graph inspired by the internal diffusion limited aggregation (IDLA) model. In both processes $n$ particles start from an arbitrary but fixed origin. Each particle performs a simple random walk…

When sand flows out of a funnel onto a surface, a three dimensional pile that is stabilized by friction grows taller as it spreads. Here we investigate an idealized two dimensional analogue: spreading of a pile of monodisperse oil droplets…

Consider a graph with a set of vertices and oriented edges connecting pairs of vertices. Each vertex is associated with a random variable and these are assumed to be independent. In this setting, suppose we wish to solve the following…

The Abelian Sandpile Model is a discrete diffusion process defined on graphs (Dhar \cite{DD90}, Dhar et al. \cite{DD95}) which serves as the standard model of self-organized criticality. The transience class of a sandpile is defined as the…

We show that for each $\lambda \in [\frac{1}{2}, 1]$, there exists a solvable group and a finitely supported measure such that the associated random walk has upper speed exponent $\lambda$.

We study the stabilization time of a wide class of processes on graphs, in which each node can only switch its state if it is motivated to do so by at least a $\frac{1+\lambda}{2}$ fraction of its neighbors, for some $0 < \lambda < 1$. Two…

Let $D(n,r)$ be a random $r$-out regular directed multigraph on the set of vertices $\{1,\ldots,n\}$. In this work, we establish that for every $r \ge 2$, there exists $\eta_r>0$ such that $\text{diam}(D(n,r))=(1+\eta_r+o(1))\log_r{n}$. Our…

Let $X$ be a weighted graph and $M$ be its adjacency, Laplacian or signless Laplacian matrix. In a continuous quantum walk on $X$, local $\epsilon$-uniform mixing occurs at vertex $u$ if the $u$th column of the matrix $U(t)=e^{itM}$ can be…

We consider a one-dimensional run-and-tumble particle, or persistent random walk, in the presence of an absorbing boundary located at the origin. After each tumbling event, which occurs at a constant rate $\gamma$, the (new) velocity of the…

We give a new proof that any candy-passing game on a graph G with at least 4|E(G)|-|V(G)| candies stabilizes. (This result was first proven in arXiv:0807.4450.) Unlike the prior literature on candy-passing games, we use methods from the…

Benjamini, Lyons and Schramm [Random Walks and Discrete Potential Theory (1999) 56-84] considered properties of an infinite graph G, and the simple random walk on it, that are preserved by random perturbations. In this paper we solve…

The paper analyzes a specific class of random walks on quotients of $X:=\text{SL}(k,{\Bbb R})/ \Gamma$ for a lattice $\Gamma$. Consider a one parameter diagonal subgroup, $\{g_t\}$, with an associated abelian expanding horosphere, $U\cong…

We study the probability that a random walk started inside a subgraph of a larger graph exits that subgraph (or, equivalently, hits the exterior boundary of the subgraph). Considering the chance a random walk started in the subgraph never…

We study analytically a simple random walk model on a one-dimensional lattice, where at each time step the walker resets to the maximum of the already visited positions (to the rightmost visited site) with a probability $r$, and with…

Activated Random Walk is a system of interacting particles which presents a phase transition and a conjectured phenomenon of self-organized criticality. In this note, we prove that, in dimension 1, in the supercritical case, when a segment…

We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski gasket ($SG$) when particles are launched from the corner vertex. In particular, the abelian sandpile…

We consider the model of random interlacements on transient graphs, which was first introduced by Sznitman [Ann. of Math. (2) (2010) 171 2039-2087] for the special case of ${\mathbb{Z}}^d$ (with $d\geq3$). In Sznitman [Ann. of Math. (2)…