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

Abstract: Fix a bounded $3$-polygon $(Ω; x_1, x_2, x_3)$ with three marked boundary points $x_1, x_2, x_3\in\partialΩ$ and suppose $(Ω^δ; x_1^δ, x_2^δ, x_3^δ)$ is an approximation of $(Ω; x_1, x_2, x_3)$ on $δ$-scaled hexagonal lattice. We consider uniform spanning tree (UST) in $Ω^δ$ with wired boundary conditions. Conditional on the event that both branches from $x_1^δ$ and $x_2^δ$ hit the boundary through $x_3^δ$, the two branches meet at a point $\mathfrak{t}^δ$ which we call trifurcation, and the union of the three branches from $x_j^δ$ to $\mathfrak{t}^δ$ 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.