Tripod in uniform spanning tree and 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 $\trifurcation^{\delta}$ which we call trifurcation, and the union of the three branches from $x_j^{\delta}$ to $\trifurcation^{\delta}$ 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$_2$. The proof relies on construction of a new observable for trifurcation in our key lemma--Lemma~3.1--where we use Fomin's formula and the geometry of the hexagonal lattice in an essential way. Interestingly, the scaling limit of the observable for trifurcation coincides with the partition function for three-sided radial $\SLE_2$. Our result gives a probabilistic interpretation of the correlation function in CFT which has conformal weights $1$ at the three boundary points and has a spinless field of weights $(1,1)$ at the bulk point.