The Abelian sandpile model on an infinite tree

Abstract

We consider the standard Abelian sandpile process on the Bethe lattice. We show the existence of the thermodynamic limit for the finite volume stationary measures and the existence of a unique infinite volume Markov process exhibiting features of self-organized criticality 1 . 1 MSC 2000: Primary-82C22; secondary-60K35.

Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

“…Due to its rich mathematical structure and tractability, the model has received substantial interest in the physics literature and in recent years in the mathematical literature as well; see the review papers [9,6] and [14].…”

“…It remains an important open problem to describe the limit in more detail, and to determine the effect of the boundary in finite volumes. Recently, infinite volume versions of the sandpile process have been constructed on the one-dimensional lattice [16], on an infinite tree [14], and for a dissipative model [15]. Unlike in these articles, we do not construct a dynamics in the limit.…”

Infinite volume limit of the Abelian sandpile model in dimensions d ≥ 3

“…Arrived at this point, we can apply the results in [18], and we obtain the following. is the generator of a stationary Markov process {η t : t ≥ 0}.…”

“…Given existence of a x , and stationarity of µ under its action, we can apply the formalism developed in [18] to construct a stationary process which is informally described as follows. Starting from a µ-typical configuration η, at each site x ∈ Z d grains are added on the event times of a Poisson process N x t with mean ϕ(x), where ϕ(x) satisfies the condition…”

“…The review papers [12,21,27] explain many points that are presented in less detail here, and may be useful for the reader.…”