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

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

2011-06-27

arxiv.org/abs/1106.5453v1

J. Stat. Mech. (2011) P10004

Physics

Condensed Matter

Statistical Mechanics

12 pages, 11 figures

Scientific paper

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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