Steady state of Stochastic Sandpile Models

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 $\le 12$, and also study the density profile in the steady state.

16 pages, 4 figures

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• Spectral properties of zero temperature dynamics in a model of a compacting granular column
• Dynamic correlations in the conserved Manna sandpile
• Directed nonabelian sandpile models on trees
• Two-component abelian sandpile models
• Combinatorial aspects of sandpile models on wheel and fan graphs
• A stochastic variant of the abelian sandpile model
• Markov Chains from Descent Operators on Combinatorial Hopf Algebras
• Uniform threshold for fixation of the stochastic sandpile model on the line