1999/02/15 by Deepak Dhar
Computer Science · Mathematics · Physics and Astronomy · #Abelian sandpile model #Boundary value problem #Combinatorics #Conjecture #Eigenvalues and eigenvectors #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Operator (biology) #Physics #Polynomial #Pure mathematics #Quantum mechanics #Statistical physics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Topological and Geometric Data Analysis #Upper and lower bounds #cond-mat.stat-mech
paper · pdf · doi:10.1016/s0378-4371(99)00149-1
12 pages. 3 eps figures. Minor revision of text. Some typographical errors fixed. Talk given at StatPhys-Calcutta III, Jan. 1999. To appear in Physica A
arxiv created 1999/02/15 · openalex publication_date 1999/08/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.