Return probability for the loop-erased random walk and mean height in the Abelian sandpile model: a proofWe are delighted to dedicate this work to Deepak Dhar on the occasion of his 60th birthday.
Poghosyan, Vahagn S; Priezzhev, Vyatcheslav B; Ruelle, Philippe; Poghosyan, Vahagn S; Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, B-1348 Louvain-La-Neuve, Belgium; Priezzhev, Vyatcheslav B; Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia; Ruelle, Philippe; Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, B-1348 Louvain-La-Neuve, Belgium
Журнал:
Journal of Statistical Mechanics: Theory and Experiment
Дата:
2011-10-01
Аннотация:
Single site height probabilities in the Abelian sandpile model, and the corresponding mean height ⟨h⟩, are directly related to the probability P<sub>ret</sub> 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<sub>ret</sub> = 5/16. We provide a rigorous proof of this conjecture by using a local monomer–dimer formulation of these questions.
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