Résumés

Mini-cours

Francesco Caravenna - On the two-dimensional KPZ and Stochastic Heat Equation via directed polymers

The Kardar-Parisi-Zhang equation (KPZ) and the multiplicative Stochastic Heat Equation (SHE) in two space dimensions are singular stochastic PDEs which lack a robust solution theory. It is standard to consider a regularized version of these equations - e.g. by convolving the noise with a smooth mollifier - and then to investigate the behavior of the regularized solution in the limit when the regularization is removed.

Crucially, the regularized solution can be interpreted as the partition function of a directed polymer in random environment, a much studied model in statistical mechanics. Building on this representation, we will show that a phase transition emerges, as we vary the disorder strength on a logarithmic scale, with an explicit critical point. In the sub-critical regime, the regularized solution exhibits so-called Edwards-Wilkinson fluctuations, i.e. after centering and rescaling it converges to an explicit Gaussian random field, solution of the additive SHE. We will finally present recent progresses in the critical regime, where many questions are still open.

These lectures are based on joint works with R. Sun and N. Zygouras.

Hugo Duminil-Copin - Critical Points of Percolation Models

In recent years, a large class of new percolation models have become tractable to mathematical analysis. These models can be constructed as classical random-graph models (like the Fortuin-Kasteleyn percolation or loop models) or they can be obtained from random functions by taking the super-level lines (like GFF or random waves). One important question  regarding these models is the computation of their critical point. In these lectures, we will review some recent developments and present some future directions of research. 

Laure Saint-Raymond - Gaz parfaits et limites de Boltzmann-Grad

Vincent Vargas - Une introduction à la théorie conforme des champs de Liouville

Exposés

Marie Albenque - Random triangulations coupled with an Ising model

Angel and Schramm proved in 2003, that uniform planar triangulations converge for the local topology. The limiting law, known as UIPT (for Uniform Infinite Planar Triangulation) has been much studied since and is now a well understood object. In this talk, I will study random triangulations with an Ising configuration sampled according to their energy and will prove that they also converge for the local topology. The limiting object turns out to be much harder to study than the UIPT and is believed to belong to another universality class at criticality. I will also present some of the very natural open questions raised by the introduction of this Ising weighted Infinite Triangulation.

This is a joint work with Laurent Ménard and Gilles Schaeffer.

Vincent Beffara - RSW without FKG

The Harris-FKG inequality is one of the most fundamental tools in statistical mechanics, and in percolation in particular; it is at the basis of many classical arguments. In particular, it allows to prove the Russo-Seymour-Welsh (RSW) estimates that states that crossing probabilities for rectangles of a given shape, for critical percolation, are bounded below uniformly in the size of the rectangle, which then leads to a rich understanding of the critical regime. I will present recent joint work with D. Gayet (Grenoble) extending the RSW results to some models for which the FKG inequality does not hold (typically, the anti-ferromagnetic Ising model at high enough temperature on the triangular lattice).

Djalil Chafai - Aspects of mean-field Gibbs measures with singular interaction

We will focus on mean-field Gibbs measures with singular interaction which appear at various places in mathematical physics. This includes Coulomb gases of random matrix theory, as well as Laughlin wave functions in fractional quantum Hall effect. For these Gibbs measures we will study quantitatively as well as asymptotically various high dimensional properties. We will also say some few words on numerical stochastic simulation, and formulate some natural open problems.

Dmitry Chelkak - Planar Ising model at criticality: state-of-the-art and perspectives

We give a brief survey of recent developments on the conformal invariance of the critical planar Ising model on ℤ² and, more generally, of the critical Z-invariant Ising model on isoradial graphs (rhombic lattices). We then introduce a new class of embeddings of general weighted planar graphs into the complex plane (s-embeddings), which paves the way to
true universality results for the critical planar Ising model.

Nicolas Curien - Critical parking on a random tree...and random planar maps!

Imagine a plane tree together with a configuration of particles (cars) at each vertex. Each car tries to park on its node, and if the latter is occupied, it moves downward towards the root trying to find an empty slot. This model has been studied recently by Bruner and Panholzer as well as Goldschmidt and Przykicki where is it shown that parking of all cars obeys a phase transition ruled by the density of cars. We study the annealed critical model of random plane tree together with a parking configuration of cars. Surprisingly this object is connected to stable looptree of parameter 3/2 and to processes encountered in the theory of random planar maps! The talk is based on ongoing work with Olivier Hénard.

Bernard Derrida - The importance of large deviations in non-equilibrium systems

In the  last two decades, major efforts were devoted to extend our undertanding of the statistical laws of fluctuations and large deviations to non-equilibrium systems. This talk will try to present some of the main recent progresses.

Jean-Baptiste Gouéré - Percolation and first passage percolation in the Boolean model

Throw in an independent and homogeneous way random balls in an Euclidean space. Denote by S the union of the balls. Here are two related questions:

1.  Are the connected components of S bounded ?
2. A walker travels at speed one outside S and at infinite speed inside S. If he takes an optimal path, is the time needed to go from the origin to a far point x asymptotically proportional to the Euclidean norm of x ?

One easily checks that if the answer to the first question (about percolation) is no, then the answer to the second question (about first passage percolation) is no. In this talk, we will investigate the converse. Based on works with Marie Théret.

Nina Holden - Natural measures on random fractals

Several fractals that arise as the scaling limit of statistical physics models come equipped with a natural measure. This measure can often be defined equivalently in multiple ways: axiomatically, via Minkowski content, or as the limit of counting measure for the discrete model. We study such measures for the Schramm-Loewner evolution, percolation pivotal points, and various fractals in the geometry of Liouville quantum gravity. Based on joint works with subsets of Bernardi, Lawler, Li, and Sun.

Arnaud Le Ny - Gibbs measures for long-range Ising models in dimensions 1 and 2

During this talk, we shall revisit old results from the 80's about one dimensional long-range polynomially decaying Ising models (sometimes called Dyson models) and describe more recent results about interface fluctuations and interface states of similar models in dimension 1 and 2. Joint works with R. Bissacot, E. Endo, A. van Enter on one hand (CMP 2018) and L. Coquille, A. Van Enter and W. Ruszel on the other hand (JSP 2018).

Cristina Toninelli - Bootstrap percolation and kinetically constrained spin models: critical time scales

Recent years have seen a great deal of progress in understanding the behavior of bootstrap percolation models, a particular class of monotone cellular automata. In the two dimensional lattice there is now a quite complete understanding of their evolution starting from a random initial condition, with a universality picture for their critical behavior. Much less is known for their non-monotone stochastic counterpart, namely kinetically constrained models (KCM). In KCM each vertex is resampled (independently) at rate one by tossing a p-coin iff it can be infected in the next step by the bootstrap model. In particular infection can also heal, hence the non-monotonicity. Besides the connection with bootstrap percolation, KCM have an interest in their own : when p→0 they display some of the most striking features of the liquid/glass transition, a major and still largely open problem in condensed matter physics. I will discuss some recent results on the characteristic time scales of KCM as p→0 and the connection with the critical behavior of the corresponding bootstrap models.

Fabio Toninelli - The domino shuffling algorithm: a (2+1)-dimensional stochastic growth model

Based on joint works with Sunil Chhita (1802.05493) and with Alexei Borodin (1806.10467)

Yvan Velenik - Asymptotics of correlations in the Ising model

I will consider the Ising model on ℤ^d. An important problem is to understand the asymptotic behavior of the covariance between two local functions f and g as the distance between their supports diverges, including corrections to the leading exponential decay.

Such questions were first tackled more than a century ago, but, up until the early 2000s, their rigorous analysis was restricted to either perturbative regimes or to the planar model with no external field. Since then, thanks to the so-called Ornstein-Zernike theory, substantial progress has been made.

I will present a survey of the known results for the Ising model on ℤ^d. I will be mainly interested in the regimes in which the correlation length is finite, that is, either when the magnetic field h is nonzero, or when h=0 and beta is different from β_c.

Wendelin Werner -TBA