The Abelian Sandpile Model on a Random Binary Tree

Journal article (2012)

Authors

F.H.J. Redig Applied Probability -

W.M. Ruszel Radboud Universiteit Nijmegen, Eindhoven University of Technology

Ellen Saada University of Paris

Research Group

Applied Probability () (TU Delft)

Phase transition Sandpile models Random binary trees

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http://resolver.tudelft.nl/uuid:844421f7-94ab-4bf5-9dac-afdcc6adfb30

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Published Date

2012

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Applied Probability

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.