Uniform spanning trees and random matrix statistics

TL;DR

This paper studies the geometry and scaling limits of branches in a conditioned uniform spanning tree on a grid, revealing connections to random matrix eigenvalues, SLE processes, and Gaussian free fields, with surprising parity-dependent behaviors.

Contribution

It provides an exact formula for the winding characteristic function, describes the scaling limit of branches as eigenvalues of a COE matrix, and develops a coupling of SLE with the Gaussian free field.

Findings

Winding behavior depends only on the parity of the number of branches.

Branches in the scaling limit match eigenvalues of a COE matrix.

Variance of the winding scales as ppa/n^2, aligning with some physics predictions.

Abstract

We consider a uniform spanning tree in a $\delta$-square grid approximation of a planar domain $\Omega$. For given integer $n\ge 2$, we condition the tree on the following $n$-arm event: we pick $n$ branches, emanating from $n$ points microscopically close to a given interior point, and condition them to connect to the boundary $\partial \Omega$ without intersecting. What can be said about the geometry of these branches? We derive an exact formula for the characteristic function of the total winding of the branches. A surprising consequence of this formula is that in the scaling limit, the behaviour of this function depends on the total number of branches $n$ only through its parity. We also describe the scaling limit of the branches. If $\Omega $ is the unit disc, then they hit the boundary (i.e., the unit circle) at random positions which coincide exactly with the eigenvalues of a random matrix of size $n$ drawn from the Circular Orthogonal Ensemble (COE, also called C$\beta$E with $\beta =1$). Furthermore, the branches converge to Loewner evolution driven by the circular Dyson Brownian motion with parameter $\beta = 4$ (i.e., $n$-sided radial SLE$_\kappa$ with $\kappa=2$). We thus verify a prediction made by Cardy in this setting. Along the way, we develop a flow-line (imaginary geometry) coupling of $n$-sided radial SLE$_\kappa$ with the Gaussian free field, which may be of independent interest. Surprisingly, we find that the variance of the corresponding field near the singularity also does not depend on the number $n\ge 2$ of curves. In contrast, the variance of the the winding of the curves behaves as $\kappa/n^2$, which agrees with the predictions from the physics literature made by Wieland and Wilson numerically, and by Duplantier and Binder using Coulomb gas methods -- but disagrees with a result of Kenyon.