Abelian sandpiles: an overview and results on certain transitive graphs

We review the Majumdar-Dhar bijection between recurrent states of the Abelian sandpile model and spanning trees. We generalize earlier results of Athreya and Jarai on the infinite volume limit of the stationary distribution of the sandpile model on Z^d, d >= 2, to a large class of graphs. This includes: (i) graphs on which the wired spanning forest is connected and has one end; (ii) transitive graphs with volume growth at least c n^5 on which all bounded harmonic functions are constant. We also extend a result of Maes, Redig and Saada on the stationary distribution of sandpiles on infinite regular trees, to arbitrary exhaustions.Comment: 44 pages. Version 2 incorporates some smaller changes. To appear in Markov Processes and Related Fields in the proceedings of the meeting: Inhomogeneous Random Systems, Stochastic Geometry and Statistical Mechanics, Institut Henri Poincare, Paris, 27 January 201

Sandpile models

This survey is an extended version of lectures given at the Cornell Probability Summer School 2013. The fundamental facts about the Abelian sandpile model on a finite graph and its connections to related models are presented. We discuss exactly computable results via Majumdar and Dhar's method. The main ideas of Priezzhev's computation of the height probabilities in 2D are also presented, including explicit error estimates involved in passing to the limit of the infinite lattice. We also discuss various questions arising on infinite graphs, such as convergence to a sandpile measure, and stabilizability of infinite configurations.Comment: 72 pages - v3 incorporates referee's comments. References closely related to the lectures were added/update

Phase transition in a sequential assignment problem on graphs

We study the following game on a finite graph $G = (V, E)$. At the start, each edge is assigned an integer $n_e \ge 0$, $n = \sum_{e \in E} n_e$. In round $t$, $1 \le t \le n$, a uniformly random vertex $v \in V$ is chosen and one of the edges $f$ incident with $v$ is selected by the player. The value assigned to $f$ is then decreased by $1$. The player wins, if the configuration $(0, \dots, 0)$ is reached; in other words, the edge values never go negative. Our main result is that there is a phase transition: as $n \to \infty$, the probability that the player wins approaches a constant $c_G > 0$ when $(n_e/n : e \in E)$ converges to a point in the interior of a certain convex set $\mathcal{R}_G$, and goes to $0$ exponentially when $(n_e/n : e \in E)$ is bounded away from $\mathcal{R}_G$. We also obtain upper bounds in the near-critical region, that is when $(n_e/n : e \in E)$ lies close to $\partial \mathcal{R}_G$. We supply quantitative error bounds in our arguments.Comment: 28 pages, 2 eps figures. Some mistakes have been corrected, and the introduction has been re-written. Minor corrections throughou

Electrical resistance of the low dimensional critical branching random walk

We show that the electrical resistance between the origin and generation n of the incipient infinite oriented branching random walk in dimensions d<6 is O(n^{1-alpha}) for some universal constant alpha>0. This answers a question of Barlow, J\'arai, Kumagai and Slade [2].Comment: 44 pages, 3 figure

Anchored burning bijections on finite and infinite graphs

Let $G$ be an infinite graph such that each tree in the wired uniform spanning forest on $G$ has one end almost surely. On such graphs $G$, we give a family of continuous, measure preserving, almost one-to-one mappings from the wired spanning forest on $G$ to recurrent sandpiles on $G$, that we call anchored burning bijections. In the special case of $\mathbb{Z}^d$, $d \ge 2$, we show how the anchored bijection, combined with Wilson's stacks of arrows construction, as well as other known results on spanning trees, yields a power law upper bound on the rate of convergence to the sandpile measure along any exhaustion of $\mathbb{Z}^d$. We discuss some open problems related to these findings.Comment: 26 pages; 1 EPS figure. Minor alterations made after comments from refere