Sandpile models : The infinite volume model , Zhang ’ s model and limiting shapes

Invariant measures and limiting shapes in sandpile models

Abelian Sandpile Models in Infinite Volume∗

Since its introduction by Bak,Tang and Wiesenfeld, the abelian sandpile dynamics has been studied extensively in finite volume. There are many problems posed by the existence of a sandpile dynamics in an infinite volume S: its invariant distribution should be the thermodynamic limit (does the latter exist?) of the invariant measure for the finite volume dynamics; the extension of the sand grain...

Limiting Shapes for a Non-Abelian Sandpile Growth Model and Related 353 Cellular Automata

We present limiting shape results for a non-abelian variant of the abelian sandpile growth model (ASGM), some of which have no analog in the ASGM. One of our limiting shapes is an octagon. In our model, mass spreads from the origin by the toppling rule in Zhang’s sandpile model. Previously, several limiting shape results have been obtained for the ASGM using abelianness and monotonicity as main...

. PR ] 1 5 Fe b 20 07 Limiting shapes for deterministic internal growth models

We study the rotor router model and two deterministic sandpile models. For the rotor router model in Z, Levine and Peres proved that the limiting shape of the growth cluster is a sphere. For the other two models, only bounds in dimension 2 are known. A unified approach for these models with a new parameter h (the initial number of particles at each site), allows to prove a number of new limitin...

Infinite volume limits of high-dimensional sandpile models

We study the Abelian sandpile model on Z. In d ≥ 3 we prove existence of the infinite volume addition operator, almost surely with respect to the infinite volume limit μ of the uniform measures on recurrent configurations. We prove the existence of a Markov process with stationary measure μ, and study ergodic properties of this process. The main techniques we use are a connection between the st...