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

TL;DR

This paper provides analytical insights into Manna's stochastic sandpile model, identifying invariants, eigenvalue relations, and bounds on recurrent configurations, and proposes a conjecture for the exact minimum particle number.

Contribution

It introduces new invariants, constructs ladder operators, and establishes bounds and a conjecture for the minimum particles in recurrent configurations, advancing understanding of the model's structure.

Findings

Eigenvalues satisfy coupled polynomial equations

Constructed nontrivial toppling invariants and ladder operators

Boundaries for the minimum number of particles in recurrent states

Abstract

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.