Work

The Abelian Sandpile Model on a Random Binary Tree

Year: 2012

Type: article

Abstract: We study the abelian sandpile model on a random binary tree. Using a transfer matrix approach introduced by Dhar and Majumdar, we prove exponential decay of correlations, and in a small supercritical region (i.e., where the branching process survives with positive probability) exponential decay of avalanche sizes. This shows a phase transition phenomenon between exponential decay and power law decay of avalanche sizes. Our main technical tools are: (1) A recursion for the ratio between the numbers of weakly and strongly allowed configurations which is proved to have a well-defined stochastic solution; (2) quenched and annealed estimates of the eigenvalues of a product of n random transfer matrices.

Source: Journal of Statistical Physics

DOI: 10.1007/s10955-012-0498-6

Authors: F. Redig, W. M. Ruszel, E. Saada