Timetable (RGMW06)

We consider Boolean percolation in dimension d. Around every point of a Poisson point process of intensity lambda, draw a ball of random radius, independently for different points. We investigate the connection probabilities in the subcritical regime and use the randomized algorithm method to prove that the phase transition in lambda is sharp. Interestingly, for this process, sharpness of the phase transition does not imply exponential decay of connection probabilities in the subcritical regime, and its meaning depends on the  law of the radii. In this talk, we will focus on this specific feature of Boolean percolation. This talk is based on a joint work with H. Duminil-Copin and A. Raoufi.

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.

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.

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.

I will discuss prospects for deriving quantum theory (Virasoro representations, spectrum and bootstrap) from the probabilistic formulation of Liouville conformal field theory.

In the third lecture of the series, I will talk about the master loop equation and gauge-string duality. Recent rigorous results will be discussed and proof sketches will be given. As in the previous lectures, many open problems will be stated.

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.

The Ricci Flow on surfaces evolves the Riemannian metric to the constant curvature metric in the same conformal class. It has a natural interpretation as an infinite dimensional gradient flow. We describe the corresponding Langevin dynamics and its relation to Liouville theory. Joint work with Hao Shen.

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.

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.

In these three lectures, I plan to describe our ongoing research program with Jason Miller and Scott Sheffield that deals with the natural interplay between Conformal Loop Ensembles and the Liouville Quantum gravity area measures. I will explain some features in the continuous world of CLE and LQG/GFF that do hopefully enlighten some of the basic features of the Quantum gravity approach to critical systems, and give a roadmap to understand the scaling limit of some decorated discrete planar maps models.

Random constraint satisfaction problems encode many interesting questions in the study of random graphs such as the chromatic and independence numbers. Ideas from statistical physics provide a detailed description of phase transitions and properties of these models. We will discuss the one step replica symmetry breaking transition that many such models undergo and the Satisfiability Threshold for the random K-SAT model.

I will present a joint work with Nathanaël Berestycki and Goirab Ray in which we prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain Markov property is a multiple of the Gaussian free field. This result holds subject only to a fourth moment assumption.

In a recent work, we have derived an almost sure multifractal boundary spectrum for SLE_kappa(rho)-processes. Finding the upper bound on the dimension is fairly standard and is done with the help of Girsanov's theorem. The lower bound on the dimension, however, is a bit more cumbersome. To find this, we use the imaginary geometry coupling of SLE and GFF to find a sufficient two-point estimate. In this talk, we will mainly focus on construction of the perfect points and the method of finding the two-point estimate.

Random constraint satisfaction problems encode many interesting questions in the study of random graphs such as the chromatic and independence numbers. Ideas from statistical physics provide a detailed description of phase transitions and properties of these models. We will discuss the one step replica symmetry breaking transition that many such models undergo and the Satisfiability Threshold for the random K-SAT model.

In these three lectures, I plan to describe our ongoing research program with Jason Miller and Scott Sheffield that deals with the natural interplay between Conformal Loop Ensembles and the Liouville Quantum gravity area measures. I will explain some features in the continuous world of CLE and LQG/GFF that do hopefully enlighten some of the basic features of the Quantum gravity approach to critical systems, and give a roadmap to understand the scaling limit of some decorated discrete planar maps models.

The Loewner energy of a simple loop on the Riemann sphere is defined to be the Dirichlet energy of its driving function which is reminiscent in the SLE theory. It was shown in a joint work with Steffen Rohde that the definition is independent of the parametrization of the loop, therefore provides a Moebius invariant quantity on free loops which vanishes only on the circles. In this talk, I will present intrinsic interpretations of the Loewner energy (without involving the iteration of conformal distortions given by the Loewner flow), using the zeta-regularizations of determinants of Laplacians and show that the class of finite energy loops coincides with the Weil-Petersson class of the universal Teichmueller space.

We consider the thick points of random walk, i.e. points where the local time is a fraction of the maximum. In two dimensions, we answer a question of Dembo, Peres, Rosen and Zeitouni and compute the number of thick points of planar random walk, assuming that the increments are symmetric and have a finite moment of order two. The proof provides a streamlined argument based on the connection to the Gaussian free field and works in a very general setting including isoradial graphs. In higher dimensions, we show that the number of thick points converges to a nondegenerate random variable and that the maximum of the local times converges to a randomly shifted Gumbel distribution.

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.

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.

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.

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.