Fundamental constants in the theory of two-dimensional uniform spanning trees

V. S. Pogosyan^ab, V. B. Priezzhev<sup>a

a</sup> Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia
^b Institute for Informatics and Automation Problems, National Academy of Sciences of Armenia, Yerevan, Armenia

Abstract: Three characteristics of two-dimensional uniform spanning trees are nontrivially related to one another{:} the average density of a sandpile, the looping constant of a square lattice, and the return probability of a loop-erased random walk. We briefly trace the long history of the discovery of their unexpected rational values.

Keywords: Abelian sandpile model, loop-erased random walk, monomer–dimer model.

Received: 28.12.2015

DOI: 10.4213/tmf9136

 English version: Theoretical and Mathematical Physics, 2016, 187:3, 952–963

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