Limiting shapes for a non-abelian sandpile growth model and related cellular automata

2010/06/25 by Anne Fey, Haiyan Liu, Fey, Anne +1
Mathematics · #37B15 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #math.CO #math.PR #msc:37B15

paper · pdf · doi:10.48550/arxiv.1006.4928

Revised version, to appear in Journal of Cellular Automata

arxiv created 2010/08/23 · arxiv updated 2010/08/24

Abstract

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 main tools. As both properties fail for our model, we use a new proof technique: in our main proof, we introduce several cellular automata to mimic our growth model.

Related

• Continuous Abelian Sandpile Model in Two Dimensional Lattice
• The Limit Shape of the Leaky Abelian Sandpile Model
• Any shape can ultimately cross information on two-dimensional abelian\n sandpile models
• A shape theorem for exploding sandpiles
• Limit shape of the leaky Abelian sandpile model with multiple layers