Schedule: Week 3 (June 15-19)

We study a directed distance on random planar maps decorated by the critical Fortuin-Kasteleyn (FK) model, which belongs to the $\gamma$-Liouville quantum gravity (LQG) universality class for $\gamma\in[\sqrt{2},2)$. The directed distance is defined as the length of the shortest path whose vertices are visited in the same order as the FK interface. We construct the Busemann function that measure directed distances to infinity along the boundary of the map, and prove that, under scaling limits, it converges to a $(2-\gamma^2/4)$-stable Lévy process.

We also obtain up-to-constants bound for the directed distance on certain finite critical FK-decorated maps in the $\gamma$-LQG universality class for $\gamma\in[\sqrt{2},2)$. For a map with $n$ edges, the shortest directed paths have length of order $n^{2/(8-\gamma^2)}$.

In the first half of the talk, I will present and try to motivate the Volume Conjecture, relating certain quantum invariants of 3-dimensional objects to their hyperbolic volumes. After this, I will explain the role of the discrete Fourier transform in the volume conjecture, and how it relates to the de Sitter/hyperbolic duality.

Toda conformal field theories are generalizations of Liouville theory for which the underlying field is vector-valued. In addition to conformal invariance, these theories are assumed in the physics literature to enjoy an enhanced level of symmetry encoded by W-algebras, that originate from the vertex algebra setting.

In this talk, we will discuss how to give a precise meaning to this assumption based on the probabilistic representation Toda theories. Namely we will characterize W-algebras in terms of holomorphic currents of Toda theories and show that Toda correlation functions satisfy Ward identities, a fundamental step in the conformal bootstrap procedure. We will also discuss some implications of this statement on solvability of Toda theories.

We study critical site percolation on the uniform infinite half-planar triangulation (UIHPT) with white–black boundary conditions. Previous work has established the convergence of percolation interface-decorated maps to $\mathrm{SLE}_6$-decorated $\sqrt{8/3}$-Liouville quantum gravity surfaces under the local GHPU topology. In this work, I will discuss the existence of the scaling limit of the loop-erasure of the percolation interface. I will also mention how, using results from ongoing work on mating-of-trees descriptions of quantum wedges, this limiting curve can be identified as $\mathrm{SLE}_{8/3}(-1;-1)$.

We discuss a novel method designed to solve SPDEs in non-translation-invariant settings using the flow approach introduced by Kupiainen and Duch. We then discuss the application of this method to the case of the Phi^4 Langevin dynamics with Neumann boundary conditions on a domain with a general curved boundary. This is joint work with Nikolay Barashkov and Majdouline Borji.

The goal of the talk will be to introduce Jackiw-Teitelboim quantum gravity, without assuming any prior knowledge of the subject. We will review standard results on the formulation of the theory on infinite-size, conformally compact geometries (the Schwarzian field theory description) together with much more recent results on the formulation of the theory on finite-size geometries. A tentative plan of the talk is as follows:

1. Brief introduction to two-dimensional quantum gravity models, from Liouville to Jackiw-Teitelboim.
2. Some physics motivation: four-dimensional near-extremal quantum black holes.
3. Conformally compact boundary conditions and the Schwarzian field theory.
4. Motivations to study finite-size geometries.
5. The lattice formulation: discretized uniform random flat metrics and non-uniform random self-overlapping polygons.
6. The continuum formulation: where JT meets Conformal Field Theory and Liouville-type techniques.
7. From finite-size to infinite-size geometries: conceptual and technical challenges.

Conformal loop ensembles (CLE) describe the scaling limits of two-dimensional critical loop models through random collections of conformally invariant fractal loops. In this talk, I will present a derivation of exact formulas for a class of disk 2-point functions for CLE. The derivation is based on the conformal field theory of SLE loop measures, which provides a natural framework for identifying the relevant fields, Ward identities, and conformal-block decompositions. Several steps in the derivation are not fully rigorous, but do not rely on guess-and-check or numerical methods. Based on joint work with Xin Sun and Baojun Wu.

We present the Brownian loop measure on hyperbolic surfaces for Brownian motion with a constant killing rate. We compute the mass of the Brownian loop measure with killing in a free homotopy class and then relate the total mass of loops in all essential homotopy classes to the Selberg zeta function when the surface is geometrically finite. As an application, we provide a probabilistic interpretation of different notions of regularized determinants of Laplacian, in both the closed and infinite-area cases.

Random curves are key objects in the study of two-dimensional critical phenomena. Arising as domain walls in spin models or level lines in height models, simple random curves admit a rich algebraic structure described by the Temperley-Lieb algebra, which manifests the \( U_q(sl_2) \) quantum group symmetry and integrability of these models.

Bifurcating random curves emerge naturally in multi-state spin models and multi-component height models, which possess rich phase diagrams with integrable points, and exhibit higher rank symmetries both at the lattice level and in the scaling limit. However, the algebraic structure underlying these bifurcating curves remains largely unexplored.

In this talk, we define the \( A_2 \) Kuperberg algebra, that describes bifurcating random curves with \( U_q(sl_3) \) symmetry. Its relevant diagrams are trivalent bipartite graphs called Kuperberg's \( A_2 \) webs. We construct explicit cellular structure for this algebra, and identify a set of local generators. These results lay the foundation for a transfer matrix formulation of bifurcating random curves, which allows for efficient numerical and algebraic exploration of conformal field theories with higher rank symmetries.

Schramm-Loewner evolution (SLE$_\kappa$) is a one-parameter family of random fractal curves that describes the scaling limits of interfaces in two-dimensional statistical mechanics models. We consider SLE$_\kappa$ for $\kappa \in (4, 8)$, which is the regime where the curve is self-intersecting but not space-filling. We show that there exists $\delta_0>0$ such that for $\kappa \in (8 - \delta_0, 8)$, the range of an SLE$_\kappa$ curve almost surely contains a topological Sierpiński carpet. Combined with a result of Ntalampekos (2021), this implies that in this parameter range, SLE$_\kappa$ is almost surely conformally non-removable, and the conformal welding problem for SLE$_\kappa$ does not have a unique solution. Our result also implies that for $\kappa \in (8 - \delta_0, 8)$, the adjacency graph of the complementary connected components of the SLE$_\kappa$ curve is disconnected. This is joint work with Zijie Zhuang (UPenn).

Schramm-Loewner evolution (SLE) loop measures are conformally invariant measures on Jordan curves in the Riemann sphere. Restriction covariance properties, involving the central charge and Brownian loop measure, provide ways to generalize loop measures to more general configurations involving multiply connected domains, Riemann surfaces, and multiple disjoint loops. Each configuration comes with a notion of Loewner energy, or potential, which is viewed as the action functional for the loop measure.

I present a case study of a two-loop generalization of SLE in the Riemann sphere. In joint work with Yan Luo, we show that the probabilistic two-loop Loewner potential is problematic as it is not lower bounded -- and present a resolution by taking into account how CFT partition functions on the annulus bounded by the two loops depend on the modulus. My understanding is that this is representative general configurations, where the interaction between loops and moduli is determined by "cocycles" of surfaces with boundary under gluing.

We discuss the quasisymmetric geometry of the Brownian sphere, also known as the Brownian map, a random metric measure space homeomorphic to the two-dimensional sphere with Hausdorff dimension $4$ almost surely. Motivated partly by geometric group theory, the quasisymmetric uniformization problem extends classical uniformization by asking to classify fractal metric spaces up to quasisymmetric equivalence. In joint work with Jason Miller, we prove that the conformal dimension of the Brownian sphere, an important quasisymmetric invariant, is almost surely equal to $2$, matching its topological dimension. We also prove that the Brownian sphere is almost surely quasisymmetrically rigid: it has no nontrivial quasisymmetric automorphisms, and two independent Brownian spheres are almost surely not quasisymmetrically equivalent.

We define a natural orientation on a spanning-tree decorated planar map whereby, roughly speaking, a path is directed if and only if edges on the path are visited in order by the contour exploration of the spanning tree. We study directed distances (lengths of shortest directed paths) with respect to this orientation.

We construct the Busemann function which measures directed distances to $\infty$ along a natural interface in the uniform infinite spanning-tree decorated map. We show that this Busemann function, re-scaled appropriately, converges in law to a $3/2$-stable Lévy process.

We also show that in a uniform spanning tree decorated map with $n$ edges, directed distances are typically of order $n^{1/3}$. Using a strong coupling argument, we deduce analogous statements for directed distances in other random planar maps in the $\sqrt 2$-Liouville quantum gravity (LQG) universality class, including uniform meandric systems and mated-CRT maps for $\gamma=\sqrt 2$. These results give the scaling dimension for a hypothetical directed version of the $\sqrt 2$-LQG metric.

Our proof strategy is inspired by work of Borga and Gwynne (2025) on directed distances in bipolar-oriented triangulations.

Based on joint work with Jacopo Borga and Ewain Gwynne.

We derive an exact formula for the probability that a Brownian path on an annulus does not disconnect the two boundary components of the annulus. The leading asymptotic behavior of this probability is governed by the disconnection exponent obtained by Lawler-Schramm-Werner (2001) using the connection to Schramm-Loewner evolution (SLE). The derivation of our formula is based on this connection and the coupling with Liouville quantum gravity (LQG). As byproducts of our proof, we obtain a precise relation between Brownian motion on a disk stopped upon hitting the boundary and the SLE loop measure on the disk; we also obtain a detailed description of the LQG surfaces cut by the outer boundary of stopped Brownian motion on an LQG disk.