The abelian sandpile model on a random binary tree

TL;DR

This paper investigates the abelian sandpile model on a random binary tree, demonstrating a phase transition between exponential and power-law decay of avalanches, using transfer matrix methods and probabilistic analysis.

Contribution

It introduces a transfer matrix approach to analyze the model on a random tree and proves a phase transition in avalanche decay behavior.

Findings

Exponential decay of correlations established.

Exponential decay of avalanche sizes in a supercritical region proven.

Identification of a phase transition between decay types.

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.