Some Results and a Conjecture for Manna's Stochastic Sandpile Model

Details Some Results and a Conjecture for Manna's Stochastic Sandpile Model Some Results and a Conjecture for Manna's Stochastic Sandpile Model

1999-02-10

arxiv.org/abs/cond-mat/9902137v2

Physics

Condensed Matter

Statistical Mechanics

12 pages. 3 eps figures. Minor revision of text. Some typographical errors fixed. Talk given at StatPhys-Calcutta III, Jan. 19

Scientific paper

10.1016/S0378-4371(99)00149-1

We present some analytical results for the stochastic sandpile model, studied earlier by Manna. In this model, the operators corresponding to particle addition at different sites commute. The eigenvalues of operators satisfy a system of coupled polynomial equations. For an L X L square, we construct a nontrivial toppling invariant, and hence a ladder operator which acting on eigenvectors of evolution operator gives new eigenvectors with different eigenvalues. For periodic boundary conditions in one direction, one more toppling invariant can be constructed. We show that there are many forbidden subconfigurations, and only an exponentially small fraction of all stable configurations are recurrent. We obtain rigorous lower and upper bounds for the minimum number of particles in a recurrent configuration, and conjecture a formula for its exact value for finite-size rectangles.

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