Steady State and Relaxation Spectrum of the Oslo Rice-pile

Abstract

We show that the one-dimensional Oslo rice-pile model is a special case of the abelian distributed processors model. The exact steady state of the model is determined. We show that the time evolution operator W for the system satisfies the equation W^{n+1} = W^n where n = L(L+1)/2 for a pile with L sites. This is used to prove that W has only one eigenvalue 1 corresponding to the steady state, and all other eigenvalues are exactly zero. Also, all connected time-dependent correlation functions in the steady state of the pile are exactly zero for time difference greater that n. Generalization to other abelian critical height models where the critical thresholds are randomly reset after each toppling is briefly discussed.

arXiv comment11 pages, latex, no figures

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• Steady state of Stochastic Sandpile Models
• Anchored advected interfaces, Oslo model, and roughness at depinning
• Sampling rare fluctuations of height in the Oslo ricepile model
• Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles