Random Conformal Geometry and Related Fields in Jeju

We consider various configuration exponents of Hamiltonian paths drawn on bicubic random planar maps. Estimates from exact finite size results are compared with predictions based on the KPZ relations, as applied to regular exponents on the hexagonal lattice. Surprisingly, a naive use of KPZ does not reproduce all the measured exponents but a certain Ansatz may possibly account for the observed discrepancies. We further study Hamiltonian cycles on various families of bicolored planar maps, which fall into two universality classes, with central charges $c = -1$ or $c = -2$. The first group comprises $p$-regular maps of fixed vertex valency $p$ greater or equal to $3$, and the second, maps of mixed vertex valencies, and a so-called rigid case. For each class, a universal configuration exponent, as well as a novel critical exponent associated with long-distance contacts along a Hamiltonian cycle are obtained from KPZ and the corresponding exponent on regular (hexagonal or square) lattices. This time, the predictions are numerically confirmed by enumeration results for $p$-regular maps with $p = 3; 4; : : : ; 7,$ and for maps with mixed valencies $(2; 3), (2; 4)$ and $(3; 4)$.
Based on joint work with Ph. Di Francesco, O. Golinelli and E. Guitter.

The double random current (DRC) model is a natural percolation model whose geometric properties are intimately related to spin correlations of the Ising model. In two dimensions, it moreover carries an integer valued height function on the graph, called the nesting field. We study the critical DRC model on bounded domains of the square lattice. We fully describe the joint scaling limit of the (primal and dual) DRC clusters and the nesting field as the lattice mesh size vanishes. We prove that the nesting field becomes the Dirichlet Gaussian free field (GFF) in this limit, and that the outer boundaries of the DRC clusters with free boundary conditions are the conformal loop ensemble with $\kappa=4$ (CLE4) coupled to that GFF. Moreover, we also show that the inner boundaries of the DRC clusters form a two-valued local set with values ${\mp 2\lambda, (2\sqrt2 \mp 2) \lambda}$ for the field restricted to a CLE4 loop with boundary value $\pm 2\lambda$. Our proof is a combination of exact solvability of the Ising model, new crossing estimates for the DRC model (which does not posses the FKG property), and a careful analysis of the structure of two-valued local sets of the continuum GFF. This is joint work with Hugo Duminil-Copin and Wei Qian.

Given a finite number of points on the sphere, which simple closed curves minimize the Loewner energy among all curves that pass through these points? We will discuss several characterizations of such curves in terms of hyperbolic geometry, conformal welding, and differential equations. This is joint work with Mario Bonk, Janne Junnila, Don Marshall and Yilin Wang.

We derive the variational formula of the Loewner driving function of a simple chord under infinitesimal quasiconformal deformation with Beltrami coefficients supported away from the chord. As an application, we obtain the first variation of the Loewner energy of a Jordan curve, defined as the Dirichlet energy of the driving function of the curve. This gives another explanation of the identity between the Loewner energy and universal Liouville action, introduced by Takhtajan and Teo. Considering the quasiconformal deformation allows us to identify the holomorphic stress-energy tensor of the Loewner energy as the Schwarzian derivative of the uniformizing conformal map of the complement of the curve, which is at the heart of many identities of the Loewner energy, e.g., with renormalized volume, a Fredholm determinant involving the Grunsky operator, etc. This is a joint work with Jinwoo Sung.

For each central charge $c\in (0,1]$, we construct a conformally invariant field which is a measurable function of the local time field $\mathcal{L}$ of the Brownian loop soup with intensity $c$ and i.i.d. signs given to each cluster. This field is canonically associated to $\mathcal{L}$, in a sense which is similar to the isomorphism theory that associates the Gaussian free field to the loop soup with critical intensity. Isomorphisms between Brownian motions and random fields were previously developed by Symanzik, Brydges-Fröhlich-Spencer, Dynkin and Le Jan in several different settings.
In a key intermediate step, we obtain the crossing exponent for the event that a cluster in the subcritical loop soup passes near a given point. Among other things, it allows us to deduce that the Minkowski gauge function of a cluster in a loop soup with intensity $c\in (0,1)$, which is $t^2 f(t)$ for some $f(t)=|\log t|^{1-c/2+o(1)}$.
I will also present several conjectures about this family of fields. This talk is based on joint works with Antoine Jego (EPFL) and Titus Lupu (CNRS).

A meandric system is a collection of loops obtained from two non-crossing perfect matchings on $\{1,\ldots,2n\},$ one drawn above the real line and the other below the real line. A meander is a meandric system with exactly one loop. We conjectured that the scaling limit of meanders (resp. meandric systems) should be described by $\sqrt{(1/3(17- \sqrt{145}))}$-Liouville quantum gravity sphere (resp. $\sqrt{2}$ -Liouville quantum gravity sphere) decorated by two Schramm-Loewner evolutions with parameter $\kappa=8$ (resp. decorated by a Schramm-Loewner evolution with parameter $\kappa=8$ and a conformal loop ensemble with parameter $\kappa=6$). In this talk, we will introduce these conjectures and discuss recent progress in studying meanders and their conjectural scaling limit. The case of meandric systems will be addressed in Park’s talk.
Based on joint works with Ewain Gwynne, Xin Sun, and Minjae Park.

In Borga's talk, the large-scale geometry of a uniformly sampled meandric system of size n is conjectured to be described by an independent triple consisting of a Liouville quantum gravity (LQG) sphere with parameter $\gamma=\sqrt2$, a Schramm-Loewner evolution (SLE) curve with parameter $\kappa=8$, and a conformal loop ensemble (CLE) with parameter $\kappa=6$. I will further discuss the geometry of uniform meandric systems and present several rigorous results consistent with this conjecture. In particular, a uniform meandric system admits macroscopic loops, and the half-plane version of the meandric system has no infinite paths. Based on joint work with Jacopo Borga and Ewain Gwynne.

I will present a few results on the regularity of the SLE trace. There are several notions of regularity such as variation and modulus of continuity. The optimal leading exponents (a.k.a. $p$-variation resp. Hölder exponent) have been previously identified by several authors. Here, we will focus on logarithmic refinements. In the case of variation regularity (which is a notion that depends only on the curve modulo parametrisation), we can prove the optimal (up to constants) result. The modulus of continuity depends on the parametrisation; for the natural parametrisation, we have the optimal (up to constants) result, whereas for the capacity parametrisation, we have a fairly good upper bound. Finally, I will comment on the regularity of the uniformising maps for SLE. Part of this talk is joint work with Nina Holden.

I will explain some recent work on the relationship between Yang-Mills theory and the theory of random surfaces.

We consider the Schramm-Loewner evolution (SLE$_\kappa$) with $\kappa=4$, the critical value of $\kappa > 0$ at or below which SLE$_\kappa$ is a simple curve and above which it is self-intersecting. We show that the range of an SLE$_4$ curve is a.s. conformally removable. Such curves arise as the conformal welding of a pair of independent critical ($\gamma=2$) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. In order to establish this result, we give a new sufficient condition for a set $X \subseteq {\mathbf C}$ to be conformally removable which applies in the case that $X$ is not necessarily the boundary of a simply connected domain. We will also describe how this theorem can be applied to obtain the conformal removability of the SLE$_\kappa$ curves for $\kappa \in (4,8)$ in the case that the adjacency graph of connected components of the complement is a.s. connected. Based on joint work with Konstantinos Kavvadias and Lukas Schoug.

We consider critical planar site percolation on the triangular lattice and derive sharp estimates on asymptotic probability of half-plane $j$-arm events for $j\geq 1$ and planar (polychromatic) $j$-arm events for $j>1$ under some conditions. These estimates greatly improve previous ones and solve (a large part of) an open question of Schramm (Proc. of ICM, 2006). This is joint work with Hang Du (PKU), Yifan Gao (CityU HK) and Zijie Zhuang (UPenn).

Conformal dimension of a set is the minimal Hausdorff dimension of its quasisymmetric image. In the talk, I will discuss the conformal dimensions of various deterministic and stochastic sets, such as Bedford-McMullen sets and self-affine Fractal Percolation Clusters. I will show that the Brownian graph is minimal, i.e. its conformal dimension is equal to 3/2, its Hausdorff dimension. The talk is based on joint works with Hrant Hakobyan (Kansas State) and Wenbo Li (Toronto).

Conformal invariance of critical lattice models in two-dimensional has been vigorously studied for decades. The first example where the conformal invariance was rigorously verified was the planar uniform spanning tree (together with loop-erased random walk), proved by Lawler, Schramm and Werner around 2000. Later, the conformal invariance was also verified for Bernoulli percolation (Smirnov 2001), level lines of Gaussian free field (Schramm-Sheffield 2009), and Ising model and FK-Ising model (Chelkak-Smirnov et al 2012). In this talk, we focus on connection probabilities of these critical lattice models in polygons with alternating boundary conditions.
This talk has two parts.

• In the first part, we consider critical random-cluster model with cluster weight $q\in (0,4)$ and give conjectural formulas for connection probabilities of multiple interfaces. The conjectural formulas are proved for $q=2$, i.e. the FK-Ising model.
• In the second part, we consider uniform spanning tree (UST) and give formulas for connection probabilities of multiple Peano curves. UST can be viewed as the limit of random-cluster model as $q$ goes to 0. Its connection probabilities turn out to be related to logarithmic CFT.

I discuss a hidden quantum group symmetry in 2D CFT with central charge less than one. It provides a systematic method for solving Belavin-Polyakov-Zamolodchikov PDE systems, and in particular for explicit computation of the asymptotics and monodromy properties of the solutions. Moreover, using a quantum Schur-Weyl duality, one can understand solution spaces of such PDE systems in a detailed way. The solutions, in turn, are useful both for CFT questions and for rigorous understanding of the connections of CFT with critical models of statistical physics and interacting SLE curves.

Liouville quantum gravity (LQG) is a natural model for a random fractal surface, and one canonical finite volume LQG surface is the LQG disk. A powerful tool in the study of LQG is conformal welding, where multiple LQG surfaces are combined into a single LQG surface, with the interfaces typically being variants of Schramm-Loewner evolutions (SLE). In this talk, we will present our recent result on conformal welding of multiple LQG disks. In particular, the LQG disks can be conformally welded together according to any given topological link pattern. The interface is multiple chordal SLE, while the random conformal structure of the marked points is encoded by the $\kappa<4$ multiple SLE pure partition function. Based on joint work with Ang and Sun.

In this talk, I will present some preliminary recent results on the Multiple SLE model with a Dyson Brownian Motion driver. Dyson Brownian motion is a very effective tool in the study the Universality of Random Matrix Ensembles. In my presentation, I will investigate this Multiple SLE model focusing on two particular cases. In the first case, the analysis of Bessel processes is effective and in the second one tools from Random Matrices are useful. Both of the works study the behaviour of the corresponding Loewner chains under perturbations of parameters. In the last part of the talk, I will present some future directions that I plan to study. The result in the first case is obtained jointly with J. Chen, and the one in the second case is part of a recent study performed jointly with A. Campbell, and K. Luh from the Random Matrix Theory community.

I will discuss the problem of constructing Gaussian multiplicative chaos measures from level sets of the thick points of the field, where the thickness measured using some fixed approximation scheme of the field. In a recent joint work with G. Lambert and C. Webb we showed convergence in distribution of the level set measures to GMC assuming only certain asymptotic behaviour of the exponential moments of the approximations used. In particular the approximations can be non-Gaussian and as an application we were able to prove a conjecture of Fyodorov and Keating on the fluctuations of the mass of thick points of the CUE characteristic polynomial.

We prove the existence of the intersection of an SLE$_\kappa(\rho)$ curve with a real interval using the standard Green’s function approach. Then we show the existence of Minkowski content of the intersection of an SLE$_\kappa(\rho)$ curve with real intervals, which further implies the existence of Minkowski content measure. SLE$_\kappa(\rho)$ bubble measures resemble SLE$_\kappa$ loop measures. The rooted SLE$_\kappa(\rho)$ measures exist for all $\kappa>0$ and $\rho>-2$, which satisfies some SLE$_\kappa(\rho)$-related domain Markov property, and is the weak limit of the usual SLE$_\kappa(\rho)$ curves with the two endpoints tending to the same point. For $\kappa\in(0,8)$ and $\rho\in ((-2)\vee(\frac\kappa 2-4),\frac\kappa 2-2)$, we derive a decomposition formula for the rooted SLE$_\kappa(\rho)$ bubble with respect to the boundary Minkowski content, and use it to further construct unrooted SLE$_\kappa(\rho)$ bubble measures.

Certain measures on fractals carry a lot of information about and provide us with ways of sampling points on said fractals. In this talk we describe how to use the GFF to construct measures on random fractals in general focus on the case of CLE in particular. We prove the existence and uniqueness of a conformally covariant measure on CLE which respects the CLE Markov property and note that it is the Minkowski content of CLE, provided that the latter exists. This talk is based mainly on joint work with Jason Miller.

We check several properties that were predicted by physicists about the correlations in the scaling limits of the critical Ising model on planar domains. Among them are the bosonization, i.e., the equality between the square of an Ising correlation and a correlation for the (compactified) Gaussian free field, the second-order PDE known as the Belavin-Polyakov-Zamolodchikov (BPZ) equations, and the structure of operator product expansions (OPE) to all orders, as predicted by conformal field theory. Joint work with Baran Bayraktaroglu, Tuomas Virtanen, and Christian Webb.

We prove the convergence of correlations between the spin, disorder, fermion, and energy fields in the planar Ising model under massive scaling limit. Massive analogues of the critical operator product expansions (OPE's) are explicitly given. The correlation functions in the limit may be bosonized and identified with correlations in the massless Sine-Gordon model. Based on joint works with Chelkak, Izyurov, Webb, and others.

During this presentation, I will cover several topics related to boundary Liouville CFT, including the current understanding of structure constants, the BPZ equation, the conformal bootstrap method, and how these concepts relate to 2D statistical models.

The mating-of-trees theory of Duplantier, Miller, and Sheffield states that a $\gamma$-LQG surface explored by an independent space-filling SLE($16/\gamma^2$) curve has an infinitely divisible law, where the relevant observables are the LQG measures in bulk and on the boundary of SLE segments. We extend this formulation to the local LQG metric within SLE cells and show that, almost surely, the $\gamma$-LQG diameter of a finite segment of the space-filling SLE($16/\gamma^2$) curve is finite if and only if $\gamma<\sqrt{8/3}$. This fact resolves a conjecture of Gwynne, Holden, and Sun on the distance exponent for mated-CRT maps. This is joint work with Yuyang Feng (University of Chicago).