Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The Green’s function on the double cover of the grid and application to the uniform spanning tree trunk

Richard W. Kenyon and David B. Wilson

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Abstract

We compute the Green’s function on the double cover of $\mathbb{Z}^{2}$, branched over a vertex or a face. We use this result to compute the local statistics of the “trunk” of the uniform spanning tree on the square lattice, i.e., the limiting probabilities of cylinder events conditional on the path connecting far away points passing through a specified edge. We also show how to compute the local statistics of large-scale triple points of the uniform spanning tree, where the trunk branches. The method reduces the problem to a dimer system with isolated monomers, and we compute the inverse Kasteleyn matrix using the Green’s function on the double cover of the square lattice. For the trunk, the probabilities of cylinder events are in $\mathbb{Q}[\sqrt{2}]$, while for the triple points the probabilities are in $\mathbb{Q}[1/\pi ]$.

Résumé

Nous calculons la fonction de Green sur le revêtement à deux feuillets de $\mathbb{Z}^{2}$, ramifié au-dessus d’un sommet ou d’une face. Nous utilisons ce résultat pour calculer les statistiques locales du « tronc » de l’arbre couvrant minimal sur le réseau carré, c’est-à-dire les probabilités limites des événements cylindriques conditionnées à ce que le chemin connectant deux sommets éloignés passe par une arête donnée. Nous montrons également comment calculer les statistiques locales des points triples à grande échelle de l’arbre couvrant minimal, où le tronc se sépare. La méthode consiste à ramener le problème à un système de dimères avec des monomères isolés, et nous calculons l’inverse de la matrice de Kasteleyn à l’aide de la fonction de Green sur le revêtement deux feuillets du réseau carré. Pour le tronc, les probabilités des événements cylindriques sont dans $\mathbb{Q}[\sqrt{2}]$, tandis que pour les points triples, les probabilités sont dans $\mathbb{Q}[1/\pi]$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 3 (2020), 1841-1868.

Dates
Received: 11 September 2018
Revised: 3 July 2019
Accepted: 16 August 2019
First available in Project Euclid: 26 June 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1593137311

Digital Object Identifier
doi:10.1214/19-AIHP1019

Mathematical Reviews number (MathSciNet)
MR4116710

Subjects
Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Keywords
Loop-erased random walk Laplacian Green’s function

Citation

Kenyon, Richard W.; Wilson, David B. The Green’s function on the double cover of the grid and application to the uniform spanning tree trunk. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 3, 1841--1868. doi:10.1214/19-AIHP1019. https://projecteuclid.org/euclid.aihp/1593137311


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