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.
Highlights
The abelian sandpile model (ASM) is a thoroughly studied model both in the physics and in the mathematics literature see e.g. [3, 4, 9, 12, 15] for recent review papers on the subject. It serves as a paradigmatic model of self-organized criticality (SOC)
The ASM is connected to several combinatorial objects such as spanning trees, graph-orientations, dimers, and it has an interesting abelian group structure
The ASM has been studied on the Bethe lattice in [5]
Summary
The abelian sandpile model (ASM) is a thoroughly studied model both in the physics and in the mathematics literature see e.g. [3, 4, 9, 12, 15] for recent review papers on the subject. We use the transfer-matrix method of [5] to express relevant quantities such as the correlation of height variables and the avalanche size distribution in terms of the eigenvalues of an ad hoc product of random matrices. This is the fundamental difference between the Bethe lattice case and the random tree, namely the fact that the transfer matrices depend randomly on the vertices and instead of having to deal with the n-th power of a simple two by two matrix, one has to control the product of n random matrices. We give quenched and annealed estimates of the eigenvalues of the product of n random matrices, which we apply in the study of correlation of height variables and avalanche sizes
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