Random Conformal Geometry and Related Fields

The first result I will describe is the existence of the (discrete) loop-erased random walk in two and three dimensions. I will then discuss issues about the scaling limit in two dimensions including the relation of the transition probability to random walks with zipper (with Christian Benes and Fredrik Viklund) and two-sided annulus SLE (with Mohammad Jahangashahi).

We show that for a class of non-thin local sets of the $2D$ GFF, satisfying a one-sided boundedness condition, the restriction of the GFF to this sets is a Minkowski content measure in gauge $|log(r)|^(1/2) r^2.$ This class of local sets includes the first passage sets of the GFF. The proof goes through the Gaussian multiplicative chaos. As a consequence, this gives, via an isomorphism theorem, the gauge to measure the size of clusters in a critical $2D$ Brownian loop-soup. This is joint work with Avelio Sepulveda (Lyon 1) and Juhan Aru (ETH Zurich).

Over the past years, the planar Gaussian free field has been conjectured, and in some cases proved, to arise as a universal scaling limit from a broad range of statistical physics models. I will discuss a recent work with Nathanaël Berestycki and Gourab Ray, showing that any random distribution satisfying conformal invariance and a certain domain Markov property must be a multiple of the Gaussian free field. This result holds subject only to a fourth moment assumption on the field.

In this talk, I will introduce a version of conformal field theory (CFT) and explain its implementations to SLE theory in a doubly connected domain. The statistical fields in these implementations are OPE families of central charge modifications of the Gaussian free field with excursion reflected/Dirichlet boundary conditions. After presenting certain equations in CFT including a version of Eguchi-Ooguri and Ward’s equations, I will outline the relation between CFT and SLE theory. By means of screening, I will explain how to find Euler integral type solutions to the parabolic partial differential equations for the annulus SLE partition functions introduced by Lawler and Zhan and present a class of SLE martingale-observables associated with these solutions. This is based on joint work with Nam-Gyu Kang and Hee-Joon Tak.

Fix $\kappa \in (0,8)$ and suppose that $\eta$ is an SLE$_\kappa$ curve in $\mathbb{H}$ from $0$ to $\infty$. We show that if $\varphi: \mathbb{H} \to \mathbb{H}$ is a homeomorphism which is conformal on $\mathbb{H} \setminus \eta$ and $\varphi(\eta)$, $\eta$ are equal in distribution then $\varphi$ is a conformal automorphism of $\mathbb{H}$. Applying this result for $\kappa=4$ establishes that the welding operation for critical ($\gamma=2$) Liouville quantum gravity (LQG) is well-defined. Applying it for $\kappa \in (4,8)$ gives a new proof that the welding of two looptrees of quantum disks to produce an SLE$_\kappa$ on top of an independent $4/\sqrt{\kappa}$-LQG surface is well-defined. These results are special cases of a more general uniqueness result which applies to any non-space-filling SLE-type curve (e.g., the exotic SLE$_\kappa^\beta(\rho)$ processes). This is a joint work with Oliver McEnteggart and Jason Miller.

The Loewner's energy of a simple loop is defined to be the Dirichlet energy of its driving function. It depends a priori on the parametrization of the simple loop. However, it was shown in a joint work with Steffen Rohde that there is no such dependence, therefore provides a Moebius invariant quantity for free loops on the Riemann sphere which vanishes only on circles. In this talk, I will present an intrinsic interpretation of the Loewner energy using the zeta-regularizations of determinants of Laplacians and show that the class of finite energy loops coincides with the Weil-Petersson class in the universal Teichmueller space.

Every finite combinatorial tree can be canonically embedded in the complex sphere as the solution to a certain conformal welding problem. We consider the properties of this embedding for large random trees. In particular we prove the existence, uniqueness and regularity of the limiting random conformal map as the number of edges goes to infinity. Joint work with Steffen Rohde.

In its most well-known form, the Loewner equation gives a correspondence between curves in the upper half-plane and continuous real functions (called the "driving function" for the equation). We consider the generalized Loewner equation, where the driving function has been replaced by a time-dependent real measure. In the first part of the talk, we investigate the delicate relationship between the driving measure and the generated hull, specifying a class of discrete random driving measures that generate hulls in the upper half-plane that are embeddings of trees. In the second part of the talk, we consider the probabilistic question of finding the scaling limit of these measures for Galton-Watons trees. We particularly focus on distributions on trees converging to the continuum random tree, and we conclude by describing a connection to the complex Burgers equation.

Benjamini and Kesten introduced in 1995 the problem of embedding infinite binary sequences into a Bernoulli percolation configuration, known as percolation of words. We give a positive answer to their Open Problem 2: for site percolation on $\mathbb{Z}^3$ with parameter $p=1/2,$ we prove that almost surely, all words can be embedded. We also discuss various extensions of this result. This talk is based on a joint work with Augusto Teixeira (IMPA) and Vincent Tassion (ETH Zurich).

Conformal invariance and critical phenomena in two-dimensional statistical physics have been active areas of research in the last few decades. This talk concerns conformally invariant random curves that should describe scaling limits of interfaces in critical lattice models. The scaling limit of the interface in critical planar lattice model with Doburshin boundary conditions (b.c.), if exists, should satisfy conformal invariance (CI) and domain Markov property (DMP). In 1999, O. Schramm introduced SLE process, and this is the only one-parameter family of random curves with CI and DMP. In this talk, we first discuss the scaling limit of a pair of interfaces in rectangle with alternating b.c. The scaling limit of such pair, if exists, should satisfy CI, DMP and symmetry (SYM). It turns out there is a two-parameter family of random curves satisfying CI, DMP, and SYM, and they are Hypergeometric SLE. Next, we will explain how to generalize to the more general setting where the notions of multiple SLEs and pure partition functions emerge. This talk is based on the joint works with V. Beffara and E. Peltola.

We study the adjacency graph of bubbles - i.e., complementary connected components - of an SLE$_{\kappa}$ curve for $\kappa \in (4,8)$, with two such bubbles considered to be adjacent if their boundaries intersect. We show that this adjacency graph is a.s. connected for $\kappa \in (4,\kappa_0]$, where $\kappa_0 \approx 5.6158$ is defined explicitly. This gives a partial answer to a problem posed by Duplantier, Miller and Sheffield (2014). Our proof in fact yields a stronger connectivity result for $\kappa \in (4,\kappa_0]$, which says that there is a Markovian way of finding a path from any fixed bubble to $\infty$. We also show that there is a (non-explicit) $\kappa_1 \in (\kappa_0, 8)$ such that this stronger condition does not hold for $\kappa \in [\kappa_1,8)$. Our proofs are based on an encoding of SLE$_\kappa$ in terms of a pair of independent $\kappa/4$-stable processes, which allows us to reduce our problem to a problem about stable processes. We will explain in the talk how this encoding comes from the theory of Liouville quantum gravity. This is joint work with Ewain Gwynne."

In this talk, we construct the quantum theory of the coupling of the Liouville action and of Mabuchi's K-energy. Both functionals play a prominent role respectively in Riemannian geometry and Kähler geometry. As an output, we obtain a path integral whose Weyl anomaly presents the standard Liouville anomaly plus an additional K-energy term. Motivations come from theoretical physics where these type of path integrals have been proposed by A. Bilal, F. Ferrari, S. Klevtsov and S. Zelditch as a model for fluctuating metrics arising when coupling (small) massive perturbations of conformal field theories to quantum gravity. Our probabilistic construction relies on a variant of Gaussian multiplicative chaos (GMC), the Derivative GMC (DGMC for short). The main technical backbone of our construction is twofold and consists in two estimates on (derivative and standard) GMC which are of independent interest. First, we show that these DGMC random variables possess negative exponential moments; second, we derive optimal small deviations estimates for GMC (associated to a recentered GFF).

We describe recent advances in the multifractality of the harmonic measure of Schramm-Loewner Evolution (SLE). A generalized notion of integral means spectrum is introduced, that involves the averaged moments of both the SLE conformal map and its derivative, and provides a unified description of the interior and exterior versions of whole-plane SLE. In the moment plane, several integrability loci exist, with four analytical forms of the spectrum separated by phase transition lines. These forms are generalized to complex moments. Based on joint works with Dmitry Belyaev, Ilia Binder, Xuan Hieu Ho, Thanh Binh Le, and Michel Zinsmeister.

Multiple SLEs are conformally invariant measures on families of curves, that naturally correspond to scaling limits of interfaces in critical planar lattice models with alternating ("generalized Dobrushin") boundary conditions (proofs of convergence existing for LERW, Ising, FK-Ising, and percolation). I briefly discuss classification of such measures via two approaches: a "global" classification by a natural cascade property, and a "local" classification as SLE variants with given partition functions. On the other hand, the partition functions should give scaling limits of connection probabilities in critical planar lattice models (proofs existing for multichordal LERWs, double-dimer pairings, and level lines of the GFF). The talk is based on joint works with Vincent Beffara (Université Grenoble Alpes, Institut Fourier), Kalle Kytölä (Aalto University), and Hao Wu (Yau Mathematical Sciences Center, Tsinghua University).

In this talk, I will present how abstract results on SDEs can be involved in the context of Loewner Differential Equation in order to study properties of the SLE traces and conformal welding homeomorphisms. In the first part of the talk, we study a characterization of the law of the SLE tip at a fixed capacity time using a sequence of radius independent SDEs. This sequence of SDE's is obtained via a random time change in the context of the backward Loewner Differential Equation driven by $\sqrt{\kappa}B_t$. In the second part, I will focus on the real Bessel SDE, both in the backward and in the forward case. Here, properties of the Bessel SDEs are applied in the context of SLE in order to study the stability in the parameter $\kappa$ of the conformal welding homeomorphisms and of the SLE traces. This is a joint project with Prof. Dmitry Belyaev and Prof. Terry Lyons.

We first study the scaling limit of local correlations in the critical Ising model in 2D simply connected domains. The leading scaling term can be computed in terms of the conformal radius and its derivative, giving further justification to the conjecture that the scaling limit of the 2D Ising model exhibits conformal symmetry. The proof relies on delicate discrete complex analysis, which naturally generalises to the near-critical (massive) scaling regime. Accordingly, we will go on to discuss ongoing work on the $n$-point and massive cases and their relevance from the viewpoint of Conformal Field Theory. Based on joint work with R. Gheissari (first part) and C. Hongler.

A $2$-SLE$_\kappa$ ($\kappa\in(0,8)$), also called bi-chordal SLE$_\kappa$, is a pair of random curves $(\eta_1,\eta_2)$ in a simply connected domain $D$ connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal SLE$_\kappa$ curve in a complement domain. A two-curve Green's function is the rescaled probability that the two curves of a $2$-SLE$_\kappa$ both get near a marked point. We prove that for an interior point $z_0\in D$, the limit $\lim_{r\to 0^+}r^{-\alpha_0} \mathbb{P}[\mbox{dist}(z_0,\eta_j)$ < $r,j=1,2]$, where $\alpha_0=\frac{(12-\kappa)(\kappa+4)}{8\kappa}$, exists. We find the convergence rate and the exact formula of this Green's function in terms of a hypergeometric function up to a multiplicative constant. For $\kappa\in(4,8)$, we also prove the convergence of $\lim_{r\to 0^+}r^{-\alpha_0} \mathbb{P}[\mbox{dist}(z_0,\eta_1\cap \eta_2)$ < $r]$, whose limit is a constant times the previous Green's function. In addition, we study the case when the marked point $z_0$ lies on a smooth boundary arc of $D$, and obtain similar results with a differnt exponent $\alpha_0$, which is $\frac 2\kappa(12-\kappa)$. To derive these results, we work on two-time-parameter stochastic processes, and use orthogonal polynomials to derive the transition density of a two-dimensional diffusion process that satisfies some system of SDEs.

Conjecturally, critical statistical mechanics in two dimensions can be described by conformal field theories (CFT). The CFT description has in particular lead to exact and correct (albeit mostly non-rigorous) predictions of critical exponents and scaling limit correlation functions in many lattice models. The main ingredient of CFT is the Virasoro algebra, accounting for the effect of infinitesimal conformal transformations on local fields. In this talk we show that an exact Virasoro algebra action exists on the probabilistic local fields of two discrete models: the discrete Gaussian free field and the critical Ising model on the square lattice. The talk is based on joint work with Clément Hongler and Fredrik Viklund.

We derive an almost sure multifractal boundary spectrum for SLE$_\kappa(\rho)$ processes, using the so-called imaginary geometry coupling between SLE and GFF to get the correlation estimate.

We prove convergence to conformally covariant scaling limits for a family of observables in the critical 2D Ising model, including spins, energies, disorders and fermions. Combined with a recent work of Hongler, Kytölä and Viklund, this yields convergence of arbitrary local fields. We also check that the limits satisfy the fusion rules (a. k. a. operator product expansions) as predicted by Conformal Field Theory. These results also yield a systematic method of proving convergence of Ising interfaces to SLE(3) variants in a general setting. This is a joint work with D. Chelkak and C. Hongler.

Starting from the restriction of a 2d Gaussian free field (GFF) to the unit circle one can define a Gaussian multiplicative chaos (GMC) measure whose density is formally given by the exponential of the GFF. In 2008 Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of this GMC. In this talk we will explain how to prove rigorously this formula by using the techniques of conformal field theory. The key observation is that the moments of the total mass of GMC on the circle are equal to one-point correlation functions of Liouville conformal field theory in the unit disk. The same techniques also allow to derive a similar result on the unit interval [0,1] (in collaboration with Tunan Zhu). Finally we will discuss applications to random matrix theory and to the asymptotics of the maximum of the GFF.