## 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.

Original language | English |
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Pages (from-to) | 653-677 |

Number of pages | 25 |

Journal | Journal of Statistical Physics |

Volume | 147 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2012 |

## Keywords

- Sandpile models
- Random binary trees
- Phase transition