The abelian sandpile model on randomly rooted graphs and self-similar groups

Abstract

The Abelian sandpile model is an archetypical model of the physical phenomenon of self-organized criticality. It is also well studied in combinatorics under the name of chip-firing games on graphs. One of the main open problems about this model is to provide rigorous mathematical explication for predictions about the values of its critical exponents, originating in physics. The model was initially defined on the cubic lattices ℤ d , but the only case where the value of some critical exponent has been established so far is the case of the infinite regular tree—the Bethe lattice. This paper is devoted to the study of the abelian sandpile model on a large class of graphs that serve as approximations to Julia sets of postcritically finite polynomials and occur naturally in the study of automorphism group actions on infinite rooted trees. While different from the square lattice, these graphs share many of its geometric properties: they are of polynomial growth, have one end, and random walks on them are recurrent. This ensures that the behaviour of sandpiles on them is quite different from that observed on the infinite tree. We compute the critical exponent for the decay of mass of sand avalanches on these graphs and prove that it is inversely proportional to the rate of polynomial growth of the graph, thus providing the first rigorous derivation of the critical exponent different from the mean-field (the tree) value.