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Timetable (RGMW06)

RGM follow up

Monday 9th July 2018 to Friday 20th July 2018

Monday 9th July 2018
09:10 to 09:45 Registration
09:45 to 09:55 Welcome from Christie Marr (INI Deputy Director)
10:00 to 11:00 Sourav Chatterjee (Stanford University)
An introduction to gauge theories for probabilists: Part I
In this lecture series, I will introduce the basic framework of quantum Yang-Mills theories and lattice gauge theories from the probabilist’s point of view. A summary of available rigorous results and open problems will be given. This will be followed by a discussion of gauge-string duality, large N lattice gauge theories, and some recent results. In the first lecture, I will introduce the physical description of quantum Yang-Mills theories. I will also briefly discuss Wilson loops, quark confinement, and perturbative expansions.
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 I
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 Jian Ding (University of Pennsylvania)
Percolation for level-sets of Gaussian free fields on metric graphs
In this talk, I will present a new result on the chemical distance for percolation of level-sets for Gaussian free fields on metric graphs in two dimensions. I will try to present a sketch of our proof, which is based on analyzing a certain exploration martingale. Time permitting, I will also discuss some other potential applications as well as related open problems. Based on joint work with Mateo Wirth.
INI 1
14:35 to 15:20 Vincent Tassion (ETH Zürich)
The phase transition for Boolean percolation
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.
INI 1
15:20 to 15:45 Afternoon Tea
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 10th July 2018
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
Wednesday 11th July 2018
09:10 to 09:55 Gilles Schaeffer (École Polytechnique)
The combinatorics of Hurwitz numbers and increasing quadrangulations
INI 1
10:00 to 11:00 Jason Miller (University of Cambridge)
Random walk on random planar maps II
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
11:00 to 11:15 Morning Coffee
11:15 to 12:15 Antti Kupiainen (University of Helsinki)
Quantum Liouville Theory
I will discuss prospects for deriving quantum theory (Virasoro representations, spectrum and bootstrap) from the probabilistic formulation of Liouville conformal field theory.




INI 1
12:15 to 13:45 Buffet Lunch at INI
13:45 to 17:00 Free Afternoon
19:30 to 22:00 Formal Dinner at Emmanuel College (Old Library)
Thursday 12th July 2018
09:10 to 09:55 Tom Hutchcroft (University of Cambridge)
An operator-theoretic approach to nonamenable percolation
I will discuss progress on and implications of the following new conjecture: For any nonamenable transitive graph G=(V,E), the matrix of connection probabilities in critical Bernoulli percolation is bounded as an operator from L^2(V) to L^2(V).
INI 1
10:00 to 11:00 Sourav Chatterjee (Stanford University)
An introduction to gauge theories for probabilists: Part III
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.
INI 1
11:00 to 11:15 Morning Coffee
11:15 to 12:15 Gregory Miermont (ENS - Lyon)
Exploring random maps: slicing, peeling and layering - 2
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
12:15 to 13:45 Buffet Lunch at INI
13:45 to 14:30 Julien Dubedat (Columbia University)
Stochastic Ricci Flow
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.
INI 1
14:35 to 15:20 Igor Kortchemski (CNRS & École polytechnique)
Condensation in critical Cauchy Bienaymé-Galton-Watson trees
We will be interested in the structure of large Bienaymé-Galton-Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index α=1. In stark contrast to the case α∈(1,2], we show that a condensation phenomenon occurs: in such trees, one vertex with macroscopic degree emerges. One of the main tools is a limit theorem for centered downwards skip-free random walks whose steps are in the domain of attraction of a Cauchy distribution, when conditioned on a late entrance in the negative real line. As an application, we study the geometry of the boundary of random planar maps in a specific regime (called non-generic of parameter 3/2) and support the conjecture that faces in Le Gall & Miermont's 3/2-stable maps are self-avoiding. This is joint work with Loïc Richier.
INI 1
15:20 to 15:45 Afternoon Tea
Friday 13th July 2018
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
Monday 16th July 2018
09:40 to 10:00 Registration for new participants
10:00 to 11:00 Wendelin Werner (ETH Zürich)
Conformal loop ensembles on Liouville quantum gravity 1
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.
INI 1
11:00 to 11:15 Morning Coffee
11:15 to 12:15 Allan Sly (Princeton University)
Phase transitions of Random Constraint Satisfaction Problems - 1
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.




INI 1
12:15 to 13:45 Buffet Lunch at INI
13:45 to 14:30 Ellen Powell (ETH Zürich)
A characterisation of the Gaussian free field
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.
INI 1
14:35 to 14:55 Thomas Budzinski (ENS - Paris); (Université Paris-Sud 11)
Simple random walk on supercritical causal maps
We study the simple random walk on supercritical causal maps, obtained from a supercritical Galton-Watson tree by adding at each level a cycle joining neighbour vertices. In particular, we will prove that, if
the underlying tree has no leaf, the simple random walk has almost sure positive speed.





INI 1
15:00 to 15:20 Lukas Schoug (KTH - Royal Institute of Technology)
A multifractal SLE_kappa(rho) boundary spectrum
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.
INI 1
15:20 to 15:45 Afternoon Tea
Tuesday 17th July 2018
09:10 to 09:55 Tim Budd (Radboud Universiteit Nijmegen)
Lattice walks & peeling of planar maps
INI 1
10:00 to 11:00 Allan Sly (Princeton University)
Phase transitions of Random Constraint Satisfaction Problems - 2
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.




INI 1
11:00 to 11:15 Morning Coffee
11:15 to 12:15 Wendelin Werner (ETH Zürich)
Conformal loop ensembles on Liouville quantum gravity 2
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.
INI 1
12:15 to 13:45 Buffet Lunch at INI
13:45 to 14:30 Yilin Wang (ETH Zürich)
Geometric descriptions of the Loewner energy
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.
INI 1
14:35 to 15:20 Marcin Lis (University of Cambridge)
Circle patterns and critical Ising models
A circle pattern is an embedding of a planar graph in which each face is inscribed in a circle. I will define and prove magnetic criticality of a large family of Ising models on planar graphs whose dual is a circle pattern. The new construction includes as a special case the critical isoradial Ising models of Baxter.
INI 1
15:20 to 15:45 Afternoon Tea
Wednesday 18th July 2018
09:10 to 09:30 Danny Nam (Princeton University); (Seoul National University)
Cutoff for the Swendsen-Wang dynamics
The Swendsen-Wang dynamics is an MCMC sampler of the Ising/Potts model, which recolors many vertices at once based on the random-cluster representation of the model. Although widely used in practice due to efficiency, the mixing time of the Swendsen-Wang dynamics is far from being well-understood, mainly because of its non-local behavior. In this talk, we prove cutoff phenomenon for the Swendsen-Wang dynamics on the lattice at high enough temperatures, meaning that the Markov chain exhibits a sharp transition from mixed€ to well-mixed.

Joint work with Allan Sly.




INI 1
09:35 to 09:55 Antoine Jego (Cambridge Centre for Analysis)
Thick Points of Random Walk and the Gaussian Free Field
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.




INI 1
10:00 to 11:00 Wendelin Werner (ETH Zürich)
Conformal loop ensembles on Liouville quantum gravity 3
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.
INI 1
11:00 to 11:15 Morning Coffee
11:15 to 12:15 Allan Sly (Princeton University)
Phase transitions of Random Constraint Satisfaction Problems - 3
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.




INI 1
12:15 to 13:45 Buffet Lunch at INI
13:45 to 17:00 Free Afternoon
Thursday 19th July 2018
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
Friday 20th July 2018
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
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons