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

TL;DR

The paper analyzes trifurcation and tripod configurations in the uniform spanning tree on hexagonal-lattice approximations of a bounded 3-polygon, proving scaling limits for both the trifurcation point and the tripod. It shows that the trifurcation is absolutely continuous with density $p(\\Omega; x_1, x_2, x_3; z)$, given by a conformally covariant expression in terms of Poisson kernels and the conformal radius, and that conditioned on the trifurcation the tripod converges to a three-sided radial SLE$_2$ in the same domain. The key bridge between the discrete model and SLE is the observable $\\mathcal{Z}_{\\mathrm{tri}}$ (the tripod partition function), which matches the limiting trifurcation density and governs the conditional SLE law; the link is established via Fomin's formula and discrete complex analysis tools. The proofs combine Fomin-determinant observables, scaling-limit results from Chelkak–Smirnov and related works, and recent development on multi-sided radial SLE to obtain tightness and convergence for the tripod. Although the hexagonal lattice is essential to the current argument, the authors expect the results to extend to a broad class of discrete approximations.

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.

Figures (8)

• Figure 1.1: Illustration for tripod and trifurcation.
• Figure 2.1: The directions of three adjacent edges of $u$ on hexagonal lattice have two possible cases: (a) Case \ref{['eqn::threeneighbors_case1']} and (b) Case \ref{['eqn::threeneighbors_case2']}. In Case \ref{['eqn::threeneighbors_case1']}, we denote by $\bigtriangleup^{\delta}(u)$ the triangle with three vertices $u+\delta\mathrm{e}^{\mathfrak{i}\pi/3}, u-\delta, u+\delta\mathrm{e}^{-\mathfrak{i}\pi/3}$ (red dots). In Case \ref{['eqn::threeneighbors_case2']}, we denote by $\bigtriangleup^{\delta}(u)$ the triangle with three vertices $u+\delta\mathrm{e}^{2\mathfrak{i}\pi/2}, u+\delta\mathrm{e}^{4\mathfrak{i}\pi/3}, u+\delta$ (red dots). Note that $\bigtriangleup^{\delta}(u)$ is the dual face of $u$ and its area is $\frac{3\sqrt{3}}{4}\delta^2$.
• Figure 3.1: In tripod, suppose $\{u_1\rightsquigarrow e_1^{\delta}, u_2\rightsquigarrow e_2^{\delta}, u_3\rightsquigarrow e_3^{\delta}\}$ as in (a). We delete two edges $\langle u, u_1\rangle$ and $\langle u, u_2\rangle$ and add two edges $\langle x_1^{\delta}, x_1^{\delta, \circ}\rangle$ and $\langle x_2^{\delta}, x_2^{\delta, \circ}\rangle$ (the two edges in red) as in (b). Such operation induces a bijection from configurations in $\mathcal{A}_1^{\delta}\cap\mathcal{A}_2^{\delta}\cap\{\mathfrak{t}^{\delta}=u\}\cap\mathcal{B}_1^{\delta}$ to configurations in $\mathcal{C}_1^{\delta}$.
• Figure 3.2: The fat black curve indicates the boundary branch $\eta_1^{\delta}$ from $x_1^{\delta,\circ}$ to $e_3^{\delta}$. The fat blue curve indicates the boundary branch $\gamma_2^{\delta}$ from $x_2^{\delta, \circ}$ to $\eta_1^{\delta}$.
• Figure 3.3: Suppose $u, u_1, u_3\in \eta_1^\delta$ and $u_2\not\in\eta_1^{\delta}$. In (a), we have $u_2\rightsquigarrow e$ for some $e \in \mathcal{E}_{13}^\partial(\Omega^\delta)$. We delete the edge $\langle u, u_3\rangle$ and add the edge $e_1^{\delta}$. Such operation induces a bijection from configurations in $\mathcal{A}_1^{\delta}\cap\{u,u_1,u_3\in \eta_1^\delta\}\cap\{u_2\rightsquigarrow e\}$ to configurations in $\{u_3\rightsquigarrow e_3^\delta\}\cap\{u_1\rightsquigarrow e_1^\delta\}\cap\{u_2\rightsquigarrow e\}\cap\{\langle u, u_1\rangle\in\mathcal{T}^\delta\}$. In (b), we have $u_2\rightsquigarrow e$ for some $e \in \mathcal{E}_{31}^\partial(\Omega^\delta)$. We delete the edge $\langle u, u_3\rangle$ and add the edge $e_1^{\delta}$. Such operation induces a bijection from configurations in $\mathcal{A}_1^{\delta}\cap\{u,u_1,u_3\in \eta_1^\delta\}\cap\{u_2\rightsquigarrow e\}$ to configurations in $\{u_3\rightsquigarrow e_1^\delta\}\cap\{u_1\rightsquigarrow e_3^\delta\}\cap\{u_2\rightsquigarrow e\}\cap\{\langle u, u_1\rangle\in\mathcal{T}^\delta\}$.

Theorems & Definitions (71)

• Theorem 1.1
• Theorem 1.3
• Proposition 1.4
• Lemma 2.1
• proof
• Lemma 2.2: ChelkakSmirnovDiscreteComplexAnalysis
• proof
• Lemma 2.3
• proof
• Lemma 2.4