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

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

We present limiting shape results for a non-abelian variant of the abelian sandpile growth model (ASGM), some of which have no parallel 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 ma...

Invariant measures and limiting shapes in sandpile models

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

We define stabilizability of an infinite volume height configuration and of a probability measure on height configurations. We show that for high enough densities, a probability measure cannot be stabilized. We also show that in some sense the thermodynamic limit of the uniform measures on the recurrent configurations of the abelian sandpile model (ASM) is a maximal element of the set of stabil...

. 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...

Stable Dynamics of Sand Automata

In this paper, we study different notions of stability of sand automata, dynamical systems inspired by sandpile models and cellular automata. First, we study the topological stability properties of equicontinuity and ultimate periodicity, proving that they are equivalent. Then, we deal with nilpotency. The classical definition for cellular automata being meaningless in that setting, we define a...