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arXiv · 1106.5453

Return probability for the loop-erased random walk and mean height in sandpile : a proof

V. S. Poghosyan·V. B. Priezzhev·P. Ruelle
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Abstract

Single site height probabilities in the Abelian sandpile model, and the corresponding mean height $ $, are directly related to the probability $P_{\rm ret}$ that a loop erased random walk passes through a nearest neighbour of the starting site (return probability). The exact values of these quantities on the square lattice have been conjectured, in particular $ = 25/8$ and $P_{\rm ret} = 5/16$. We provide a rigourous proof of this conjecture by using a {\it local} monomer-dimer formulation of these questions.

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V. S. Poghosyan, V. B. Priezzhev, P. Ruelle. 2011-06-27. Return probability for the loop-erased random walk and mean height in sandpile : a proof. https://doi.org/10.1088/1742-5468%2F2011%2F10%2Fp10004

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