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

TL;DR

This paper rigorously proves the conjectured values of the mean height and return probability in the Abelian sandpile model on a square lattice using a local monomer-dimer formulation.

Contribution

It provides a rigorous proof of the conjectured mean height and return probability in the sandpile model, connecting these to loop-erased random walks.

Findings

Confirmed mean height <h> = 25/8

Confirmed return probability P_ret = 5/16

Established a local monomer-dimer formulation for these quantities

Abstract

Single site height probabilities in the Abelian sandpile model, and the corresponding mean height $<h>$, 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 $<h> = 25/8$ 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.