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

Abstract

Single site height probabilities in the Abelian sandpile model, and the corresponding mean height $<h>$, are directly related to the probability $P_{\mathrm{ret}}$ 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 $<h>=25/8$ and $P_{\mathrm{ret}}=5/16$. We provide a rigourous proof of this conjecture by using a {\it local} monomer-dimer formulation of these questions.

12 pages, 11 figures

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• Wind on the boundary for the Abelian sandpile model
• Boundary monomers in the dimer model
• Vacancy localization in the square dimer model
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• Numerical Study of the Correspondence Between the Dissipative and Fixed Energy Abelian Sandpile Models
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• Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle
• Grassmannian representation of the two-dimensional monomer-dimer model
• A Loop Reversibility and Subdiffusion of the Rotor-Router Walk
• Sandpile models in the large
• Sandpile probabilities on triangular and hexagonal lattices
• Correlations in the $n\to 0$ limit of the dense O(n) loop model
• Multipoint correlators in the Abelian sandpile model