Stochastic Sandpiles with Uniform Toppling Rule on the Line

TL;DR

This paper analyzes a stochastic sandpile model with a uniform toppling rule on the integer line, deriving the exact stationary distribution and demonstrating the existence of an infinite volume limit.

Contribution

It provides an exact calculation of the stationary distribution for a specific stochastic sandpile model and proves the convergence to a trivial infinite volume limit.

Findings

Exact stationary distribution on finite subsets

Infinite volume limit is the full configuration

Analysis of mass transfer during stabilization

Abstract

We consider the stochastic sandpile model with uniform toppling rule on the integer line. During a uniform toppling, with probability $1/3$ one particle is sent to the right of the toppled vertex, with probability $1/3$ one particle is sent to the left, and with probability $1/3$ two particles are sent out, one to the right and one to the left. We calculate exactly the stationary distribution of the stochastic sandpile Markov chain with this toppling rule on finite, connected subsets of the integers, and show that the infinite volume limit exists and is equal to the Dirac measure of the full configuration. For this end, we analyze where the excess mass leaves the system, when stabilizing the full configuration plus one additional particle on finite, connected subsets of the integers.