Return probability for the loop-erased random walk and mean height in the Abeliansandpile model: a proof**We are delighted to dedicate this work to Deepak Dhar on the occasion of his 60thbirthday.

Abstract

Single site height probabilities in the Abelian sandpile model, and the corresponding mean height⟨h⟩, are directly relatedto the probability Pret that a loop-erased random walk passes through a nearest neighbourof the starting site (return probability). The exact values of thesequantities on the square lattice have been conjectured, in particular⟨h⟩ = 25/8 andPret = 5/16. We provide a rigorous proof of this conjecture by using a local monomer–dimerformulation of these questions.