Translator Disclaimer
January 2020 Dimers and imaginary geometry
Nathanaël Berestycki, Benoȋt Laslier, Gourab Ray
Ann. Probab. 48(1): 1-52 (January 2020). DOI: 10.1214/18-AOP1326

Abstract

We show that the winding of the branches in a uniform spanning tree on a planar graph converge in the limit of fine mesh size to a Gaussian free field. The result holds assuming only convergence of simple random walk to Brownian motion and a Russo–Seymour–Welsh type crossing estimate, thereby establishing a strong form of universality. As an application, we prove universality of the fluctuations of the height function associated to the dimer model, in several situations.

The proof relies on a connection to imaginary geometry, where the scaling limit of a uniform spanning tree is viewed as a set of flow lines associated to a Gaussian free field. In particular, we obtain an explicit construction of the a.s. unique Gaussian free field coupled to a continuum uniform spanning tree in this way, which is of independent interest.

Citation

Download Citation

Nathanaël Berestycki. Benoȋt Laslier. Gourab Ray. "Dimers and imaginary geometry." Ann. Probab. 48 (1) 1 - 52, January 2020. https://doi.org/10.1214/18-AOP1326

Information

Received: 1 November 2016; Revised: 1 August 2018; Published: January 2020
First available in Project Euclid: 25 March 2020

zbMATH: 07206752
MathSciNet: MR4079430
Digital Object Identifier: 10.1214/18-AOP1326

Subjects:
Primary: 60B05, 60B99

Rights: Copyright © 2020 Institute of Mathematical Statistics

JOURNAL ARTICLE
52 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.48 • No. 1 • January 2020
Back to Top