Inverse avalanches in the Abelian sandpile model

Abstract

We define and study the inverse of particle addition process in the Abelian sandpile model. We show how to obtain the unique recurrent configuration corresponding to a single particle deletion by a sequence of operations called inverse avalanches. We study the probability distribution of sl, the number of "untopplings" in the first inverse avalanche. For a square lattice, we determine Prob (sl ) [7]. If from a spanning tree we delete a bond at random, the probability distribution of the number of sites disconnected is known to have power law tails. In two dimensions, this probability varies as s "~, where s is the number of sites disconnected [11].In this paper we show that the exponent in the ASM case that corresponds to deleting an edge in the spanning tree problem is related to the process of removing

Theoretical studies of self-organized criticality

“…In the Potts model formulation these will be the Oðq 2 Þ terms in the partition function in Eq. (39). It is sometimes convenient to work in the ensemble in which all waves are given equal weight in determining avalanche statistics.…”

“…Dhar and Manna showed that the distribution of the number of sites toppled in the last wave can be related to number of sites disconnected if a bond is deleted at random from a random spanning tree [39]. The latter is expressible in terms of the chemical exponent for spanning trees.…”

Introduction to the sandpile model

“…(25,27,30). Using the Kirchhoff's theorem one can determine, in principle, all of the correlations between different branches of spanning trees.…”

Deterministic Abelian Sandpile Models and Patterns

“…in [68] as hole addition then followed by hole avalanches in order to reach a stable configuration. Also inverse avalanches has been introduced before, in [69], to get back to the recurrent configuration after the deletion of a particle in a configuration of the ASM.…”