Dimers and Imaginary geometry

TL;DR

This paper demonstrates that the winding of branches in uniform spanning trees on planar graphs converges to a Gaussian free field, establishing universality of height function fluctuations in dimer models via imaginary geometry.

Contribution

It introduces a robust approach connecting uniform spanning trees to imaginary geometry, showing convergence to Gaussian free fields under minimal assumptions.

Findings

Winding of spanning tree branches converges to Gaussian free field.

Universality of height function fluctuations in dimer models.

Approach relies on minimal assumptions, not exact solvability.

Abstract

We present a general result which shows 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 true assuming only convergence of simple random walk to Brownian motion and a Russo-Seymour-Welsh type crossing estimate. As an application, we prove universality of the fluctuations of the height function associated to the dimer model, in several situations. This includes the case of lozenge tilings with boundary conditions lying in a plane, and Temperleyan domains in isoradial graphs (recovering a recent result of Li). The robustness of our approach, which is a key novelty of this paper, comes from the fact that the exactly solvable nature of the model plays only a minor role in the analysis. Instead, we rely on a connection to imaginary geometry, where the limit of a uniform spanning tree is viewed as a set of flow lines associated to a Gaussian free field.