Abelian avalanches and Tutte polynomials

Abstract

We introduce a class of deterministic lattice models of failure, Abelian avalanche (AA) models, with continuous phase variables, similar to discrete Abelian sandpile (ASP) models. We investigate analytically the structure of the phase space and statistical properties of avalanches in these models. We show that the distributions of avalanches in AA and ASP models with the same redistribution matrix and loading rate are identical. For an AA model on a graph, statistics of avalanches is linked to Tutte polynomials associated with this graph and its subgraphs. In the general case, statistics of avalanches is linked to an analog of a Tutte polynomial defined for any symmetric matrix.