Abelian Avalanches and Tutte Polynomials

We introduce a class of deterministic lattice models of failure, Abelian avalanche (AA) models, with continuous phase variables, similar to discrete Abelian sandpile (ASP) models. We investigate analytically the structure of the phase space and statistical properties of avalanches in these models. We show that the distributions of avalanches in AA and ASP models with the same redistribution matrix and loading rate are identical. For AA model on a graph, statistics of avalanches is linked to Tutte polynomials associated with this graph and its subgraphs. In general case, statistics of avalanches is linked to an analogue of a Tutte polynomial de ned for any symmetric matrix. Introduction. Di erent cellular automaton models of failure (sand piles, avalanches, forest res, etc.), starting with Bak, Tang and Wiesenfeld (BTW) [1], were introduced in connection with the concept of self-organized criticality [2]. Traditionally, all of these models are considered on uniform cubic lattices of di erent dimensions. Recently Dhar [3] suggested a generalization of the BTW model with an arbitrary (modulo some natural sign restrictions) matrix of redistribution of accumulated particles during an avalanche. An important property of this Abelian sand pile (ASP) model is the presence of an Abelian (commutative) group governing its dynamics. Abelian sandpiles were studied in [4], and one special case is treated in [5]. In a non-dissipative case ( P j ij = 0; for all i) an avalanche in the ASP model coincides with a chipring game on a graph [6] where is a Laplace matrix of the underlying graph. Abelian Avalanches Page 2 Another class of lattice models of failure, slider block models introduced in [7] and studied in [8], as well as models [9-13] which are equivalent to quasistatic block models, have continuous time and some quantity which accumulates and is redistributed at lattice sites. This quantity is called the slope, height, stress or energy by di erent authors. In slider block models it corresponds to force [11]. We use the term height as in [3]. We introduce here a class of deterministic lattice models with continuous time and height values at the sites of the lattice, and with an arbitrary redistribution matrix. For a symmetric matrix, these models are equivalent to arbitrarily interconnected slider block systems. One of these models, which in the case of a uniform lattice coincides with models studied in [10] and in [13] (as series case a), is characterized by the same Abelian property as ASP models. We call this the Abelian avalanche (AA) model. The stationary behavior of the AA model is periodic or quasiperiodic, depending on the loading rate vector. We show however that the distribution of avalanches for a discrete, stochastic ASP model is identical to the distribution of avalanches for an arbitrary quasiperiodic trajectory (or to its average over all periodic trajectories) of a continuous, deterministic AA model with the same redistribution matrix and loading rate. For the AA model on a graph, the combinatorial structure of the phase space and the corresponding statistics of avalanches is described in terms of the invariants of the graph and its subgraphs called \Tutte polynomials" [14]. In the general case, the same is true for an analogue of a Tutte polynomial de ned for any symmetric matrix. In the rst section, we introduce di erent types of avalanche models. In the second section, we investigate the properties of AA models. In the third section, we study the structure of the set of recurrent con gurations and derive analytic formulas for the mean number of avalanches in the AA model. Some of our results are new also for ASP models. In the fourth section, we establish the equivalence of distributions of avalanches for AA and ASP models. In the fth section, we describe the structure of the phase space for AA models on a graph in terms of Tutte polynomials. In the sixth section, we describe the Abelian Avalanches Page 3 distribution of avalanches in the AA model in terms of an analogue of a Tutte polynomial for an arbitrary symmetric matrix. The proofs of the di erent statements are given in the Appendix. 1. Avalanche models. Let V be a nite set of N elements (sites), and let be a N N real matrix with indices in V , with the following properties: ii > 0; for all i; ij 0; for all i 6= j; (1)