RGM follow up
Monday 9th July 2018 to Friday 20th 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 YangMills 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 gaugestring duality, large N lattice gauge theories, and some recent results. In the first lecture, I will introduce the physical description of quantum YangMills 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 matedCRT map, a random planar map model defined out of the continuum treemating 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 levelsets of Gaussian free fields on metric graphs
In this talk, I will present a new result on the chemical distance for percolation of levelsets 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. DuminilCopin
and A. Raoufi. 
INI 1  
15:20 to 15:45  Afternoon Tea  
17:00 to 18:00  Welcome Wine Reception at INI 
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 welldefined. 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 welldefined. These results are special cases of a more general uniqueness result which applies to any nonspacefilling SLEtype 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 09:55 
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 matedCRT map, a random planar map model defined out of the continuum treemating 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) 
09:10 to 09:55 
Tom Hutchcroft (University of Cambridge) An operatortheoretic 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 gaugestring 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 BienayméGaltonWatson trees
We will be interested in the structure of large BienayméGaltonWatson 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 skipfree 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 nongeneric of parameter 3/2) and support the conjecture that faces in Le Gall & Miermont's 3/2stable maps are selfavoiding. This is joint work with Loïc Richier.

INI 1  
15:20 to 15:45  Afternoon Tea 
09:10 to 09:30 
Cyril Marzouk (Université ParisSud 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 configurationlike 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 GromovHausdorffProkhorov sense. This model covers that of pangulations when all the integers are equal to some p,
which we can allow to vary with n, without constraint; it also applies to socalled 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 matedCRT map, a random planar map model defined out of the continuum treemating 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) Yaglomtype limit theorems for branching Brownian motion with absorption
We consider onedimensional
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 Yaglomtype 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 
09:40 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 KSAT 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 ParisSud 11) Simple random walk on supercritical causal maps
We study the simple random walk on supercritical causal maps,
obtained
from a supercritical GaltonWatson 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
twopoint estimate. In this talk, we will mainly focus on construction of the
perfect points and the method of finding the twopoint estimate.

INI 1  
15:20 to 15:45  Afternoon Tea 
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 KSAT 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 zetaregularizations of determinants of Laplacians and show that the class of finite energy loops coincides with the WeilPetersson 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 
09:10 to 09:30 
Danny Nam (Princeton University); (Seoul National University) Cutoff for the SwendsenWang dynamics
The SwendsenWang dynamics is an MCMC sampler of the Ising/Potts
model,
which
recolors many vertices at once based on the randomcluster representation
of the model. Although widely used in practice due to
efficiency, the mixing time
of the SwendsenWang dynamics is far from being
wellunderstood, mainly
because of its nonlocal behavior.
In this talk, we prove cutoff phenomenon for the
SwendsenWang dynamics
on the lattice at high enough temperatures, meaning that
the Markov chain
exhibits a sharp transition from mixed to wellmixed. 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 KSAT model. 
INI 1  
12:15 to 13:45  Buffet Lunch at INI  
13:45 to 17:00  Free Afternoon 
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 nonconventional 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 ParisEst Créteil (UPEC)) The ZDirac and massive Laplacian operators in the Zinvariant 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 graphdistance 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 DingZeitouniZhang (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 spanningtree 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 spanningtree 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 FyodorovBouchaud 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 DavidKupiainenRhodesVargas. If time permits we will also discuss the connections with the quantum sphere and the quantum disk of the DuplantierMillerSheffield approach to Liouville quantum gravity.

INI 1  
10:00 to 11:00 
Nicolas Curien (Université ParisSud 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 ParisEst) Towards quantum Kähler geometry
We propose a natural framework for probabilistic Kähler geometry on a onedimensional complex manifold based on a path integral involving the Liouville action and the Mabuchi Kenergy. 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 Kenergy 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 Kenergy and produce a quantum Mabuchi Kenergy: 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 spanningtreeweighted 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 wellunderstood mathematically in any environment. We
consider external
DLA on an infinite spanningtreeweighted 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 spanningtree weighted map.
Our proof is based
on the fact that the complement of an external DLA cluster
on a spanningtree
weighted map is a spanningtree 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 logcorrelated 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 logcorrelated
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
KupiainenRhodesVargas and for the GMC
on the unit circle the FyodorovBouchaud formula has been
recently proven by Remy. In
this talk we will present additional results on GMC
measures associated to a logcorrelated
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 logsingularities 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 