Abelian Sandpile Model on Randomly Rooted Graphs and Self-Similar Groups

Michel Matter, Tatiana Nagnibeda

Published 2011-05-20, updated 2012-10-31Version 4

The main result of this paper is a rigorous proof of criticality and an explicit computation of critical exponents for the decay of avalanches in the Abelian sandpile model (ASM) on a large family of infinite graphs. We begin by introducing the notion of criticality of the ASM for limits of finite graphs in local convergence, which naturally leads to the question about criticality of the ASM in the random weak limit. Our main technical ingredient is a sufficient condition for almost sure criticality of the ASM on sequences of finite cacti (i.e., separable graphs whose blocks are cycles or single edges) under the assumption that the random weak limit is almost surely 1-ended. Examples that allow explicit computations of the critical exponents come from actions of finitely generated groups on regular rooted trees, by automorphisms. Restricting the action to the consecutive levels of the tree defines a sequence of finite graphs whose limits in the local convergence are orbital Schreier graphs for the action of the group on the boundary of the tree. In the case of iterated monodromy groups of complex polynomials, these graphs are cacti, and we show that for 1-ended ones, the critical exponent for the decay of the mass of avalanches depends on the growth of the graph. The well-known Basilica group related to $z^2-1$ gives rise to uncountably many 4-regular one-ended graphs of quadratic growth with the critical exponent for the mass of avalanches equal to 1; as well as uncountably many new non-critical examples of quadratic growth (thus not quasi-isometric to $\mathbb{Z}$). Another iterated monodromy group that we consider provides uncountably many graphs with the critical exponent equal to $2\log 2/\log 3 >1$. Finally, we also exhibit graphs of polynomial growth with arbitrarily small critical exponent.