The Abelian sandpile model on a random binary tree
type: Item
instance of: scholarly article
title: The Abelian sandpile model on a random binary tree
zbMATH Open document ID: 1269.82024
author: F. Redig
author: Wioletta Ruszel
author: Ellen Saada
published in: Journal of Statistical Physics
publication date: 2012-07-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.5131
review text: The Abelian sandpile model (ASM) is being discussed both in mathematics and physics under different names. In physics, it is known as the paradigmatic model of self-organized criticality (SOC), while, in mathematics, the ASM is known, for example, as the integrate-and-fire model having applications in neuroscience. The ASM has been studied both on the Bethe lattice (that is, a root-less binary tree) and on random graphs. There are fundamental differences between the treatments of the ASM on Bethe lattices and on random trees. The reviewed paper presents almost a textbook quality treatment of the ASM on random binary trees. It involves the application of random matrices. While on Bethe lattices, it involves the application of products of two-by-two transfer matrices so that the results involve \(n\)-th powers of such matrices, in the case of random binary trees, one has to deal with the products of \(n\) random matrices. The crucial quantity in the last case is the characteristic ratio. It is the ratio between the numbers of weakly and strongly allowed configurations. The authors (a) develop the recursion relation between the numbers of weakly and strongly allowed configurations which is proved to have a well-defined stochastic solution and, (b) obtain quenched and annealed estimates of the eigenvalues of a product of \(n\) random transfer matrices. This allows them to analyse in some detail likely phase transitions in the ASM.
reviewed by: Arkady L. Kholodenko
Mathematics Subject Classification ID: 82B26
Mathematics Subject Classification ID: 82B27
zbMATH DE Number: 6062496
zbMATH Keywords: sandpile models
zbMATH Keywords: random binary trees
zbMATH Keywords: phase transition
zbMATH Keywords: self-organized criticality
MaRDI profile type: Publication
OpenAlex ID: W3102022721
arXiv ID: 1202.5131
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DOI: 10.1007/S10955-012-0498-6
description: scientific article; zbMATH DE number 6062496