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.

We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on Z d × Z + . In dimensions d > 6, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infinite cluster is 4 3 , and thereby prove a version of the Alexander-Orbach conjecture in this setting. The proof divides into two parts. One part establishes general estimates for simple random walk on an arbitrary infinite random graph, given suitable bounds on volume and effective resistance for the random graph. A second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d > 6, by extending results about critical oriented percolation obtained previously via the lace expansion.

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Abstract:We study the Abelian sandpile model on Z d . 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 statistics of waves and uniform two-component spanning trees and results on the uniform spanning forest measure on Z d .

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