Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

Abstract

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and AlgorithmsInfinite volume limit for the stationary distribution of Abelian sandpile models Siva R. Athreya, Antal A. Járai REPORT PNA-E0304 DECEMBER 8, 2003CWI is the National Research Institute for Mathematics and Computer Science. It is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics.CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms.

Sandpiles on the Square Lattice

“…The point p0, 0q is designated sink and is special. Each non-sink point on the torus has a sand allocation (2) σ : T m ztp0, 0qu Ñ Z ě0 .…”

“…Theorems 2 and 3 are concerned with the sandpile Markov chain on the discrete torus T m , whose stationary distribution is uniform on the (finite) set of recurrent states. These finite Markov chains are related to sandpiles on the infinite grid Z 2 by theorems of [2,24]. Athreya and Járai [2] proved that the restriction of a uniform recurrent sandpile on the d-dimensional cube r´m, ms d X Z d to any fixed finite subset of Z d converges in law as m Ñ 8.…”

“…These finite Markov chains are related to sandpiles on the infinite grid Z 2 by theorems of [2,24]. Athreya and Járai [2] proved that the restriction of a uniform recurrent sandpile on the d-dimensional cube r´m, ms d X Z d to any fixed finite subset of Z d converges in law as m Ñ 8. Hence there is a limiting measure µ on recurrent sandpiles on Z d .…”

Mean-field avalanche size exponent for sandpiles on Galton–Watson trees

“…There are various interesting quantities studied, for example the avalanche size or diameter distribution depending on the underlying graph [JL93, DM90,BHJ17], the toppling durations, infinite-volume models [AJ04,MRS02], continuous height analogues [JRS15] etc.…”

Asymptotic Height Distribution in High-Dimensional Sandpiles

“…The measures P L have a weak limit P = lim L→∞ P L [2], and hence p(i) = lim L→∞ p L (i) exist, i = 0, . .…”