# Timetable (RGMW06)

 09:10 to 09:55 Nina Holden (Massachusetts Institute of Technology)Cardy embedding of uniform triangulations The Cardy embedding is a discrete conformal embedding for random planar maps which is based on percolation observables. We present a series of works in progress with the goal of showing convergence of uniform triangulations to $\sqrt{8/3}$-LQG under this embedding. The project is a collaboration with Xin Sun, and also based on our joint works with Bernardi, Garban, Gwynne, Lawler, Li, and Sepulveda. INI 1 10:00 to 11:00 Gregory Miermont (ENS - Lyon)Exploring random maps: slicing, peeling and layering - 1 The combinatorial theory of maps, or graphs on surfaces, is rich of many different approaches (recursive decompositions, algebraic approches, matrix integrals, bijective approaches) which often have probabilistic counterparts that are of interest when one wants to study geometric aspects of random maps. In these lectures, I will review parts of this theory by focusing on three different decompositions of maps, namely, the slice decomposition, the peeling process, and the decomposition in layers, and by showing how these decompositions can be used to give access to quite different geometric properties of random maps. INI 1 11:00 to 11:15 Morning Coffee 11:15 to 12:15 Sourav Chatterjee (Stanford University)An introduction to gauge theories for probabilists: Part II In the second lecture of the series, I will introduce lattice gauge theories. A brief review of available rigorous results and open problems will be given. INI 1 12:15 to 13:45 Buffet Lunch at INI 13:45 to 14:30 Ofer Zeitouni (Weizmann Institute of Science); (New York University)On the Liouville heat kernel and Liouville graph distance (joint with Ding and Zhang) INI 1 14:35 to 15:20 Wei Qian (University of Cambridge)Uniqueness of the welding problem for SLE and LQG 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 $\varphi \colon \mathbb{H} \to \mathbb{H}$ 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. INI 1 15:20 to 15:45 Afternoon Tea
 09:10 to 09:30 Cyril Marzouk (Université Paris-Sud 11)Geometry of large random planar maps with a prescribed degree sequence I will discuss some recent progress and still ongoing work about the scaling limit of the following configuration-like model on random planar maps: for every integer n, we are given n deterministic (even) integers and we sample a planar map uniformly at random amongst those maps with n faces and these prescribed degrees. Under a no macroscopic face' assumption, these maps converge in distribution after suitable scaling towards the celebrated Brownian map, in the Gromov-Hausdorff-Prokhorov sense. This model covers that of p-angulations when all the integers are equal to some p, which we can allow to vary with n, without constraint; it also applies to so-called Boltzmann random maps and yields a CLT for planar maps. INI 1 09:35 to 09:55 Joonas Turunen (University of Helsinki)Critical Ising model on random triangulations of the disk: enumeration and limits In this talk, I consider Boltzmann random triangulations coupled to the Ising model on their faces, under Dobrushin boundary conditions and at the critical point. First, the partition function is computed and the perimeter exponent shown to be 7/3 instead of the exponent 5/2 for uniform triangulations. Then, I sketch the  construction of the local limit in distribution when the two components of the Dobrushin boundary tend to infinity one after the other, using the peeling process along an Ising interface. In particular, the main interface in the local limit touches the (infinite) boundary almost surely only finitely many times, a behavior opposite to that of the Bernoulli percolation on uniform maps. Some scaling limits closely related to the perimeters of clusters are also discussed. This is based on a joint work with Linxiao Chen. INI 1 10:00 to 11:00 Gregory Miermont (ENS - Lyon)Exploring random maps: slicing, peeling and layering - 3 The combinatorial theory of maps, or graphs on surfaces, is rich of many different approaches (recursive decompositions, algebraic approches, matrix integrals, bijective approaches) which often have probabilistic counterparts that are of interest when one wants to study geometric aspects of random maps. In these lectures, I will review parts of this theory by focusing on three different decompositions of maps, namely, the slice decomposition, the peeling process, and the decomposition in layers, and by showing how these decompositions can be used to give access to quite different geometric properties of random maps. INI 1 11:00 to 11:15 Morning Coffee 11:15 to 12:15 Jason Miller (University of Cambridge)Random walk on random planar maps III We will describe some recent developments on the study of random walks on random planar maps. We first review the continuum constructions from Liouville quantum gravity as a mating of trees. We will then explain how one can analyze the behavior of random walk on the mated-CRT map, a random planar map model defined out of the continuum tree-mating constructions. Finally, we will explain how these results can be transferred to a wide variety of discrete random planar map models. This is based on joint works with Bertrand Duplantier, Ewain Gwynne, and Scott Sheffield. INI 1 12:15 to 13:45 Buffet Lunch at INI 13:45 to 14:30 Eveliina Peltola (Université de Genève)Multiple SLEs, discrete interfaces, and crossing probabilities 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. I discuss classification of these measures and how the convergence for discrete interfaces in many models is obtained as a consequence. When viewed as measures with total mass, the multiple SLEs can also be related to probabilities of crossing events in lattice models. The talk is based on joint works with Hao Wu (Yau Mathematical Sciences Center, Tsinghua University) and Vincent Beffara (Université Grenoble Alpes, Institut Fourier). INI 1 14:35 to 15:20 Jason Schweinsberg (University of California, San Diego)Yaglom-type limit theorems for branching Brownian motion with absorption We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a critical drift towards the origin so that the process eventually goes extinct with probability one. We establish precise asymptotics for the probability that the process survives for a large time t, improving upon a result of Kesten (1978) and Berestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, which also improves upon results of Kesten (1978). An important tool in the proofs of these results is the convergence of branching Brownian motion with absorption to a continuous state branching process. INI 1
 09:10 to 09:55 Perla Sousi (University of Cambridge)Capacity of random walk and Wiener sausage in 4 dimensions In four dimensions we prove a non-conventional CLT for the capacity of the range of simple random walk and a strong law of large numbers for the capacity of the Wiener sausage. This is joint work with Amine Asselah and Bruno Schapira. INI 1 10:00 to 11:00 Vincent Vargas (ENS - Paris)The semiclassical limit of Liouville conformal field theory INI 1 11:00 to 11:15 Morning Coffee 11:15 to 12:15 Beatrice de Tiliere (Université Paris-Est Créteil (UPEC))The Z-Dirac and massive Laplacian operators in the Z-invariant Ising model INI 1 12:15 to 13:45 Buffet Lunch at INI 13:45 to 14:30 Ewain Gwynne (Massachusetts Institute of Technology)The fractal dimension of Liouville quantum gravity: monotonicity, universality, and bounds We show that for each $\gamma \in (0,2)$, there is an exponent $d_\gamma > 2$, the fractal dimension of $\gamma$-Liouville quantum gravity (LQG)", which describes the ball volume growth exponent for certain random planar maps in the $\gamma$-LQG universality class, the graph-distance displacement exponent for random walk on these random planar maps, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of $\gamma$-LQG distances such as Liouville graph distance and Liouville first passage percolation. This builds on work of Ding-Zeitouni-Zhang (2018). We also show that $d_\gamma$ is a continuous, strictly increasing function of $\gamma$ and prove upper and lower bounds for $d_\gamma$ which in some cases greatly improve on previously known bounds for the aforementioned exponents. For example, for $\gamma=\sqrt 2$ (which corresponds to spanning-tree weighted planar maps) our bounds give $3.4641 \leq d_{\sqrt 2} \leq 3.63299$ and in the limiting case we get $4.77485 \leq \lim_{\gamma\rightarrow 2^-} d_\gamma \leq 4.89898$. Based on joint works with Jian Ding, Nina Holden, Tom Hutchcroft, Jason Miller, and Xin Sun. INI 1 14:35 to 15:20 Adrien Kassel (ENS - Lyon); (CNRS (Centre national de la recherche scientifique))Quantum spanning forests I will introduce a new integrable statistical physics model, which we call a quantum spanning forest (QSF) and which provides a probabilistic framework for studying spanning-tree like structures coupled to a connection (with holonomies taking values in a unitary group of arbritrary rank), both on finite and infinite graphs. I will explain that QSFs form a special case of a new general family of probability measures which we call determinantal subspace processes (DSP), and for which we develop an independent theory. Finally, I will describe some relationships between QSFs and holonomies of random walk and the covariant Gaussian free field via the study of electrical networks with holonomy. This is joint work with Thierry Lévy. INI 1 15:20 to 15:45 Afternoon Tea
 09:10 to 09:55 Guillaume Remy (CNRS - Ecole Normale Superieure Paris)Exact formulas on Gaussian multiplicative chaos and Liouville theory We will present recent progress that has been made to prove exact formulas on the Gaussian multiplicative chaos (GMC) measures. We will give the law of the total mass of the GMC measure on the unit circle (the Fyodorov-Bouchaud formula) and on the unit interval (in collaboration with T. Zhu). The techniques of proof come from the link between GMC and Liouville conformal field theory studied by David-Kupiainen-Rhodes-Vargas. If time permits we will also discuss the connections with the quantum sphere and the quantum disk of the Duplantier-Miller-Sheffield approach to Liouville quantum gravity. INI 1 10:00 to 11:00 Nicolas Curien (Université Paris-Sud 11)Random stable maps : geometry and percolation Random stable maps are discrete random Boltzmann maps with large faces that are conjecturally linked to the CLE. We review some recent results on the geometry of such graphs and their duals, and on the behavior of Bernoulli percolations on these objects. The phenomenons that appear are the analogs of those we encoutered (or conjectured) for the Euclidean CLE. In particular, the critical bond percolation process creates a duality between the dense and dilute phase of random stable maps. The talk is based on joint works with Timothy Budd, Cyril Marzouk and Loïc Richier. INI 1 11:00 to 11:15 Morning Coffee 11:15 to 12:15 Remi Rhodes (Université Paris-Est)Towards quantum Kähler geometry We propose a natural framework for probabilistic Kähler geometry on a one-dimensional complex manifold based on a path integral involving the Liouville action and the Mabuchi K-energy. Both functionals play an important role respectively in Riemannian geometry (in the case of surfaces) and Kähler geometry. The Weyl anomaly of this path integral, which encodes the way it reacts to changes of background geometry, displays the standard Liouville anomaly plus an additional K-energy term. Motivations come from theoretical physics where these type of path integrals arise as a model for fluctuating metrics on surfaces when coupling (small) massive perturbations of conformal field theories to quantum gravity as advocated by A. Bilal, F. Ferrari, S. Klevtsov and S. Zelditch in a series of physics papers. Interestingly, our computations show that quantum corrections perturb the classical Mabuchi K-energy and produce a quantum Mabuchi K-energy: this type of correction is reminiscent of the quantum Liouville theory. Our construction is probabilistic and relies on a variant of Gaussian multiplicative chaos (GMC), the Derivative GMC (DGMC for short). The technical backbone of our construction consists in two estimates on (derivative and standard) GMC which are of independent interest in probability theory. Firstly, we show that these DGMC random variables possess negative exponential moments and secondly we derive optimal small deviations estimates for the GMC associated with a recentered Gaussian Free Field. INI 1 12:15 to 13:45 Buffet Lunch at INI 13:45 to 14:05 Joshua Pfeffer (Massachusetts Institute of Technology)External DLA on a spanning-tree-weighted random planar map External diffusion limited aggregation (DLA) is a widely studied subject in the physics literature, with many manifestations in nature; but it is not well-understood mathematically in any environment. We consider external DLA on an infinite spanning-tree-weighted random planar map. We prove that the growth exponent for the external diameter of the DLA cluster exists and is equal to $2/d _{\sqrt{2}}$, where $d_{\sqrt{2}}$ denotes the `fractal dimension of $\sqrt{2}$-Liouville quantum gravity (LQG)''---or, equivalently, the ball volume growth exponent for the spanning-tree weighted map. Our proof is based on the fact that the complement of an external DLA cluster on a spanning-tree weighted map is a spanning-tree weighted map with boundary, which allows us to reduce our problem to proving certain estimates for distances in random planar maps with boundary. This is joint work with Ewain Gwynne. INI 1 14:10 to 14:30 Tunan Zhu (ENS - Paris)Distribution of gaussian multiplicative chaos on the unit interval Starting from a log-correlated field one can define by a standard regularization technique the associated Gaussian multiplicative chaos (GMC) measure with density formally given by the exponential of the log-correlated field. Very recently exact formulas have been obtained for specific GMC measures. On the Riemann sphere a proof of the celebrated DOZZ formula has been given by Kupiainen-Rhodes-Vargas and for the GMC on the unit circle the Fyodorov-Bouchaud formula has been recently proven by Remy. In this talk we will present additional results on GMC measures associated to a log-correlated field on the unit interval [0,1]. We will present a very general formula for the real moments of the total mass of GMC with log-singularities in 0 and 1. This proves a set of conjectures given by Fyodorov, Le Doussal, Rosso and Ostrovsky. As a corollary, this gives the distribution of the total mass. INI 1