Phase transition in a sequential assignment problem on graphs

We study the following sequential assignment problem on a finite graph G = (V ,E). Each edge e ∈ E starts with an integer value n e ≥ 0, and we write n =∑ e∈En e. At time t, 1 ≤ t ≤ n, a uniformly random vertex v ∈ V is generated, and one of the edges f incident with v must be selected. The value of f is then decreased by 1. There is a unit final reward if the configuration (0, . . . , 0) is reached. Our main result is that there is a phase transition: as n←∞, the expected reward under the optimal policy approaches a constant c G &gt; 0 when (n e/n : e ∈ E) converges to a point in the interior of a certain convex set R G, and goes to 0 exponentially when (n e/n : e ∈ E) is bounded away from R G. We also obtain estimates in the near-critical region, that is when (n e/n : e ∈ E) lies close to ∂R G. We supply quantitative error bounds in our arguments. </p

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

Infinite volume limit for the stationary distribution of Abelian sandpile models

We study the stationary distribution of the standard Abelian sandpile model in the box Λn = [-n, n]d ∩ Zd for d≥ 2. We show that as n→ ∞, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in Zd. This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on Zd. In the case d > 4, we also make use of Wilson’s method

Mean-field avalanche size exponent for sandpiles on Galton–Watson trees

We show that in Abelian sandpiles on infinite Galton–Watson trees, the probability that the total avalanche has more than t topplings decays as t- 1 / 2. We prove both quenched and annealed bounds, under suitable moment conditions. Our proofs are based on an analysis of the conductance martingale of Morris (Probab Theory Relat Fields 125:259–265, 2003), that was previously used by Lyons et al. (Electron J Probab 13(58):1702–1725, 2008) to study uniform spanning forests on Zd, d≥ 3 , and other transient graphs.Applied Probabilit