Existence and uniqueness of the stationary measure in the continuous Abelian sandpile

TL;DR

This paper investigates the stationary measures of a continuous Abelian sandpile model on finite subsets of Z^d, proving existence, uniqueness, and convergence properties of the invariant measure under various conditions.

Contribution

It establishes the invariance of the uniform measure on allowed configurations and proves its uniqueness for different addition interval scenarios, using coupling and ergodic theory methods.

Findings

The uniform measure is invariant under the sandpile dynamics.

Convergence to the invariant measure occurs when addition is random within an interval.

When addition is deterministic and rational, the process does not converge, but the measure remains unique.

Abstract

Let \Lambda be a finite subset of Z^d. We study the following sandpile model on \Lambda. The height at any given vertex x of \Lambda is a positive real number, and additions are uniformly distributed on some interval [a,b], which is a subset of [0,1]. The threshold value is 1; when the height at a given vertex exceeds 1, it topples, that is, its height is reduced by 1, and the heights of all its neighbours in \Lambda increase by 1/2d. We first establish that the uniform measure \mu on the so called "allowed configurations" is invariant under the dynamics. When a < b, we show with coupling ideas that starting from any initial configuration of heights, the process converges in distribution to \mu, which therefore is the unique invariant measure for the process. When a = b, that is, when the addition amount is non-random, and a is rational, it is still the case that \mu is the unique invariant probability measure, but in this case we use random ergodic theory to prove this; this proof proceeds in a very different way. Indeed, the coupling approach cannot work in this case since we also show the somewhat surprising fact that when a = b is rational, the process does not converge in distribution at all starting from any initial configuration.