Uniform spanning trees and random matrix statistics

TL;DR

This work analyzes a uniform spanning tree conditioned on an n-arm event in planar domains, deriving an exact lattice-law for the total winding and unveiling a scaling-limit that ties the branch hitting distribution to the Circular Orthogonal Ensemble and the evolving curves to n-sided radial SLE driven by circular Dyson Brownian motion. It proves convergence of discrete UST branches to a multi-curve SLE_2 with β=4 in the disk, and characterizes the hitting-point density via COE statistics, verifying Cardy's prediction in this setting. A flow-line coupling between these n-sided SLE curves and a Gaussian free field with additive monodromy is constructed, addressing a paradox about winding versus height-function variance and resolving it through branch-cut contributions. Overall, the paper links discrete UST geometry to random-matrix theory and imaginary-geometry, offering exact determinant/dimer-loop-soup identities and a robust coupling framework for multi-curve conformal geometry.

Abstract

We consider a uniform spanning tree in a $δ$-square grid approximation of a planar domain $Ω$. 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 $\partial Ω$ 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 $Ω$ 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$β$E with $β=1$). Furthermore, the branches converge to Loewner evolution driven by the circular Dyson Brownian motion with parameter $β= 4$ (i.e., $n$-sided radial SLE$_κ$ with $κ=2$). 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$_κ$ 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 $κ/n^2$, 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.

Figures (2)

• Figure 1.1: Uniform spanning tree on the graph $\Omega^\delta$ (gray), with wired (resp. free) boundary conditions on $\partial_{\hbox{out}}\Omega^\delta$ (resp. $\partial_{\hbox{in}}\Omega^\delta$), conditioned on the event $E^\delta_{\boldsymbol{x}}$, case $n=2$. The components containing the special points $x_1^\delta$ and $x_2^\delta$ are in blue. Other components are in red. All components of the tree are oriented towards the wired boundary $\partial_{\hbox{out}}\Omega^\delta$. The two disjoint oriented paths $\gamma_1^\delta$ and $\gamma_2^\delta$, from $x_1^\delta$ and $x_2^\delta$ respectively to $\partial_{\hbox{out}}\Omega^\delta$, have been highlighted in dashed blue. In the right panel, we superimpose the corresponding dual spanning tree on the planar dual lattice (in green). Its components (three in this example) are naturally oriented towards the inner boundary.
• Figure 1.2: The dimer configuration associated to $\mathcal{T}^\delta$ and its dual. The underlying tree and color coding are identical to those in Figure \ref{['F:UST']}. In the right panel, we show the associated height function. The conventions for the height are indicated at the top and right of this picture (black vertices are either crosses or black dots; all other vertices are white). By convention, the top right corner has height zero and the height increases by $\pm 1$ across edges without dimers (and by $\pm 3$ otherwise). The height function is only well defined after introducing a branch cut $\zeta$ (in red), which entails an additional jump of $n+1$ (times 4 with this normalisation, so $+12$ in this example).

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Remark 1.8
• Theorem 2.1: Wilson's algorithm Wilson:Generating_random_spanning_trees_more_quickly_than_cover_time
• Theorem 2.2: Fomin's formula Fomin:LERW_and_total_positivity