Friday, February 24, 2012

1202.5131 (Frank Redig et al.)

The abelian sandpile model on a random binary tree    [PDF]

Frank Redig, Ellen Saada, Wioletta Ruszel
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 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.
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