Steady state of Stochastic Sandpile Models

TL;DR

This paper analyzes the steady state of stochastic abelian sandpile models by examining their algebraic structure, providing exact solutions for small systems and insights into configuration probabilities and density profiles.

Contribution

It introduces a method to determine steady states of stochastic sandpile models using their algebraic properties, including eigenvalues and Jordan block structures.

Findings

Exact steady state for 1D systems of size ≤12

Determined configuration probabilities in the steady state

Analyzed density profiles in the steady state

Abstract

We study the steady state of the abelian sandpile models with stochastic toppling rules. The particle addition operators commute with each other, but in general these operators need not be diagonalizable. We use their abelian algebra to determine their eigenvalues, and the Jordan block structure. These are then used to determine the probability of different configurations in the steady state. We illustrate this procedure by explicitly determining the numerically exact steady state for a one dimensional example, for systems of size $\le12$, and also study the density profile in the steady state.