The Abelian sandpile model on a random binary tree
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.
scientific article; zbMATH DE number 6062496