Infinite volume limit for the stationary distribution of Abelian sandpile models

Antal A. Jarai, Siva R. Athreya

Research output: Contribution to journalArticlepeer-review

22 Citations (Scopus)

Abstract

We study the stationary distribution of the standard Abelian sandpile model in the box Λn = [-n, n] d ∩ ℤ d for d≥ 2. We show that as n→ ∞, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in ℤ d . This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on ℤ d . In the case d > 4, we also make use of Wilson’s method.
Original languageEnglish
Pages (from-to)197-213
Number of pages17
JournalCommunications in Mathematical Physics
Volume249
Issue number1
DOIs
Publication statusPublished - 2004

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