Three-Sided Radial SLE₂: Theory & Applications

• Three-sided radial SLE₂ is a probability measure on triplets of non-intersecting curves connecting distinct boundary points to a common interior target, defined by conformal invariance and a strong resampling property.
• Its evolution follows a multi-curve radial Loewner equation with driving angles governed by circular Dyson Brownian motion at β=2, linking the process to the quantum Calogero–Sutherland system.
• It emerges as the scaling limit of discrete models like the UST tripod and interfaces with LQG and Gaussian Free Field constructions, offering insights into random planar geometry.

Three-sided radial SLE$_2$ (three-radial SLE$_2$) is the conformally invariant probability measure on triplets of non-intersecting simple curves in a simply-connected planar domain, each connecting a distinct boundary point to a common interior target, with defining properties of conformal covariance, a precise resampling Markov property, and explicit connections to physical observables in statistical mechanics and quantum gravity. This process arises as the scaling limit of the tripod structure in the uniform spanning tree (UST), as a canonical coupling with the Gaussian Free Field (GFF), and as the conformal welding interface in Liouville quantum gravity (LQG). The partition function and driving process are characterized by the integrable structure of the quantum Calogero-Sutherland (CS) system and the circular Dyson Brownian motion at %%%%2%%%%. It represents the first nontrivial instance ($N=3$) of the general radial multiple SLE$_\kappa$ theory.

1. Definition and Characterization

Three-sided radial SLE$_2$ is defined on a simply-connected domain $\Omega \subset \mathbb{C}$ with three marked boundary points $x_1, x_2, x_3 \in \partial\Omega$ and interior point $z \in \Omega$. Let $X(\Omega; x_1, x_2, x_3; z)$ denote the set of triples $(\gamma_1, \gamma_2, \gamma_3)$ of non-intersecting simple curves with $\gamma_j$ from $x_j$ to $z$, meeting only at $z$.

The law $\mathbb{P}^3$ on $X(\Omega; x_1, x_2, x_3; z)$ is recursively specified:

1. Sample $\gamma_3$ as a radial SLE$_2(2,2)$ from $x_3$ to $z$ with force points at $x_1, x_2$.
2. Conditioned on $\gamma_3$, sample $\gamma_1$ from $x_1$ to $z$ in $\Omega \setminus \gamma_3$ as chordal SLE$_2(2)$ with a force point at $x_2$.
3. Given $(\gamma_3, \gamma_1)$, sample $\gamma_2$ from $x_2$ to $z$ in $\Omega \setminus (\gamma_3 \cup \gamma_1)$ as ordinary chordal SLE$_2$.

Key features:

• Each marginal curve is itself a radial SLE$_2$ with force points at the other boundary marks.
• The law is uniquely determined by conformal invariance and a resampling (strong Markov) property: conditionally on any two curves, the third is SLE$_2$ in the remaining domain (Ding et al., 14 Nov 2025, Huang et al., 26 Sep 2025, Makarov et al., 20 May 2025).
• All three curves are transient and jointly almost surely non-intersecting except at the common interior endpoint.

2. Loewner Description and Driving Functions

In the unit disk $\mathbb{D}$, with $x_j$ mapped to $e^{i\theta_j}$ and $z=0$, the evolution is governed by the multi-curve radial Loewner equation. Let $W_j(t) = e^{i\theta_j(t)}$: $$\partial_t g_t(z) = -g_t(z) \sum_{j=1}^3 \frac{g_t(z) + W_j(t)}{g_t(z) - W_j(t)}\,,\quad g_0(z) = z$$ The driving angles $(\theta_1, \theta_2, \theta_3)$ solve the system

$$d\theta_j(t) = \sqrt{2}\,dB_j(t) + \sum_{k \neq j} 2 \cot\left(\tfrac{\theta_j(t) - \theta_k(t)}{2}\right)dt$$

where $(B_1,B_2,B_3)$ are independent standard Brownian motions. This is the circular Dyson Brownian motion at $\beta=2$, corresponding to the eigenvalue process of the unitary ensemble (Katori et al., 19 Aug 2025, Makarov et al., 20 May 2025). The SDE ensures absolute continuity to independent SLE$_2$ up to the collision time, with no collisions almost surely for $\kappa \leq 4$.

The process is time-parametrized using radial capacity: $g_t'(0) = e^t$ gives the “common” time, while a “multi-time” parametrization is possible where each curve grows at its own radial capacity (Huang et al., 26 Sep 2025).

3. Partition Functions and Calogero–Sutherland Connection

The law of three-sided radial SLE$_2$ depends on a positive partition function $Z(\Omega; x_1, x_2, x_3; z)$ that serves as a weighting for the driving SDE drifts and prescribes boundary densities: $$Z(\Omega; x_1, x_2, x_3; z) = \frac{\left[P(\Omega; z, x_1) P(\Omega; z, x_2) P(\Omega; z, x_3)\right]^2}{CR(\Omega; z)^2 \cdot \sqrt{P(\Omega; x_1,x_2) P(\Omega; x_2,x_3) P(\Omega; x_3,x_1)}}$$ where $P(\Omega;z,x)$ is the Poisson kernel (boundary-to-interior), $CR(\Omega; z)$ is the conformal radius, and $P(\Omega;x,y)$ is the boundary Poisson kernel (Ding et al., 14 Nov 2025). This partition function is conformally covariant: under $\psi:\Omega \to \Omega'$,

$$Z(\Omega; x_1, x_2, x_3; z) = |\psi'(z)|^2 \prod_{j=1}^3 |\psi'(x_j)| Z(\Omega';\psi(x_1),\psi(x_2),\psi(x_3); \psi(z))$$

In disk coordinates, $$Z(\theta_1,\theta_2,\theta_3) = \prod_{i<j} |2\sin\frac{\theta_i-\theta_j}{2}|$$ is also the pure (electric, non-magnetic) Coulomb-gas correlator and an eigenfunction (ground state) for the quantum Calogero–Sutherland Hamiltonian with $\beta=4$: $$H = -\tfrac12 \sum_{j=1}^3 \partial_{\theta_j}^2 + \tfrac12 \sum_{i<j} \frac{1}{\sin^2 \left(\frac{\theta_i - \theta_j}{2}\right)}$$ with eigenvalue $E=1$ (Makarov et al., 20 May 2025). The null-state PDEs for $\psi(\theta)$ are

$$\mathcal{L}_j[\psi] = \left(\partial_j^2 + \sum_{k\neq j} \cot\Bigl(\frac{\theta_k - \theta_j}{2}\Bigr)\partial_k - 2 \sum_{k\neq j} \frac{1}{4\sin^2((\theta_k - \theta_j)/2)}\right) \psi = 0$$

and the solution space is three-dimensional, corresponding to pure link patterns.

4. Conformal Welding Construction in Liouville Quantum Gravity

Three-sided radial SLE$_2$ has a canonical geometric realization via conformal welding in LQG. Let $\gamma=\sqrt{2}$ ($\kappa=2$), and consider a quantum triangle $(\mathbb{H}, \phi; \infty, 0, 1)$ with weight triple $(W_1,W_2,W_3)$ and boundary quantum lengths $(\ell_1, \ell_2, \ell_3)$. When $W_1=W_2=W, W_3=2$, and $\ell_1=\ell_2=\ell$, welding the two equal-length arcs yields a quantum disk with one bulk insertion and one boundary insertion, and the welding seam is an independent radial SLE$_2(0;W-2)$ curve (Ang et al., 29 Nov 2024).

For $W=2$, this yields an SLE$_2(0;0)$—that is, ordinary three-sided radial SLE$_2$—on the standard weight-2 LQG disk. The coupling to imaginary geometry is explicit: the interface is a flow line of the GFF plus a deterministic additive term $$h_{\alpha,\beta} = h + \alpha\,\arg(\cdot)+\beta\log|\cdot|$$, with parameters determined by the force-point configuration.

5. Scaling Limit from Discrete Models and Uniform Spanning Tree

Three-sided radial SLE$_2$ arises as the scaling limit of discrete random geometry models, most notably the tripod observable in the critical uniform spanning tree (UST) (Ding et al., 14 Nov 2025). In a wired UST on a lattice domain approximating $(\Omega;x_1,x_2,x_3)$:

• Condition on the event that branches from $x_1^\delta$ and $x_2^\delta$ both reach $x_3^\delta$ via a common interior vertex $t^\delta$ (the “trifurcation”).
• The three branches from $x_j^\delta$ to $t^\delta$ converge (as $\delta\to0$) to three-sided radial SLE$_2$ branches with $t$ as target.
• The scaling limit of the trifurcation density is $p(z)=Z(\Omega;x_1,x_2,x_3;z)/\int_\Omega Z$.

Wilson’s algorithm and Fomin’s determinantal formula underlie the proof, connecting the discrete Poisson kernels to the continuum partition function and establishing absolute continuity of the scaling limit in terms of explicit boundary observables.

6. Resampling, Boundary Perturbation, and Further Properties

Three-sided radial SLE$_2$ enjoys a strong resampling property: conditionally on any two branches, the law of the third is that of a chordal SLE$_2$ in the complementary component with appropriate parameters. In particular:

• The domain Markov property holds simultaneously for all branches (multi-curve Markovianity) (Huang et al., 26 Sep 2025).
• The law is invariant under reparameterizations and local perturbations of the boundary, transforming with exponent $c=-2$ (central charge) (Huang et al., 26 Sep 2025).
• The model admits spiraling (magnetic charge) generalizations, realized by introducing a drift in the angular coordinates, but the canonical case has zero spiral parameter.

In the coupling with GFF, each SLE branch corresponds to a flow line, and the process is characterized by the partition function and the covariance structure of the GFF with prescribed jumps.

7. Connections to Hydrodynamic Limits, Quantum Integrability, and Outlook

Three-sided radial SLE$_2$ is the first nontrivial finite-$N$ instance ($N=3$) of the radial multiple SLE family, unified under the Dyson Brownian motion with $\beta = 8/\kappa$ and integrable structures from the quantum Calogero-Sutherland hierarchy (Katori et al., 19 Aug 2025, Makarov et al., 20 May 2025). In the hydrodynamic limit ($N \to \infty$), the empirical process of driving angles converges to the inviscid complex Burgers equation, with partition functions and martingales linked to conformal field theory block functions.

This framework provides a rigorous link between integrable stochastic processes, conformally invariant scaling limits, and geometric constructions in random planar maps and LQG. The scaling limit construction is universal for a broad class of discrete models with appropriate symmetry and boundary conditions.

References:

• (Ang et al., 29 Nov 2024) Radial conformal welding in Liouville quantum gravity
• (Makarov et al., 20 May 2025) Multiple radial SLE($\kappa$) and quantum Calogero-Sutherland system
• (Ding et al., 14 Nov 2025) Tripod in uniform spanning tree and three-sided radial SLE$_2$
• (Katori et al., 19 Aug 2025) Coupling of radial multiple SLE with Gaussian free field, and the hydrodynamic limit
• (Huang et al., 26 Sep 2025) Multiradial SLE with spiral: resampling property and boundary perturbation