The abelian sandpile model on a random binary tree

Abstract

We study the abelian sandpile model on a random binary tree. Using a transfer matrix approach introduced by Dhar & 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.

30 pages, 6 figures

References in corpus (1)

• Dynamical synapses causing self-organized criticality in neural networks

Cited by in corpus (2)

• Sandpiles Subjected to Sinusoidal Drive
• Non-criticality criteria for Abelian sandpile models with sources and sinks