Tripod in uniform spanning tree and three-sided radial SLE$_2$

TL;DR

This paper studies the scaling limit of a special tripod structure in uniform spanning trees on a hexagonal lattice, showing it converges to a three-sided radial SLE$_2$ with explicit distribution properties.

Contribution

It introduces the concept of trifurcation in USTs and proves its convergence to three-sided radial SLE$_2$, providing explicit density and distribution results.

Findings

Distribution of trifurcation is absolutely continuous with explicit density.

Conditional law of the tripod given trifurcation is three-sided radial SLE$_2$.

Observable for trifurcation matches the partition function for three-sided radial SLE$_2$.

Abstract

Fix a bounded $3$-polygon $(\Omega; x_1, x_2, x_3)$ with three marked boundary points $x_1, x_2, x_3\in\partial\Omega$ and suppose $(\Omega^{\delta}; x_1^{\delta}, x_2^{\delta}, x_3^{\delta})$ is an approximation of $(\Omega; x_1, x_2, x_3)$ on $\delta$-scaled hexagonal lattice. We consider uniform spanning tree (UST) in $\Omega^{\delta}$ with wired boundary conditions. Conditional on the event that both branches from $x_1^{\delta}$ and $x_2^{\delta}$ hit the boundary through $x_3^{\delta}$, the two branches meet at a point $\mathfrak{t}^{\delta}$ which we call trifurcation, and the union of the three branches from $x_j^{\delta}$ to $\mathfrak{t}^{\delta}$ form a tripod in the UST. We compute the scaling limit of the tripod: the distribution of trifurcation is absolutely continuous with respect to Lebesgue measure with explicit density; given the trifurcation, the conditional law of the tripod is three-sided radial SLE$_2$. Interestingly, the scaling limit of the observable for trifurcation coincides with the partition function for three-sided radial SLE$_2$. The proof for the distribution of the trifurcation relies on Fomin's formula [Fom01] and tools from [CS11, CW21]. The proof of the convergence to three-sided radial SLE$_2$ relies on tools developped recently from [HPW25]. We believe the conclusion is true for a large family of discrete lattice approximations, however, our proof uses the geometry of the hexagonal lattice in an essential way.