Antoine Jego (Wien), Janne Junnila (Lausanne), Ellen Powell (Durham), Avelio Sepulveda (Lyon), Vincent Vargas (Paris), Christian Webb (Aalto), Ofer Zeitouni (Rehovot).

Antoine Jego (Wien): Thick points of random walk and multiplicative chaos.

The study of thick points of planar random walk, where the walk goes back unusually often, goes back to a famous paper of Erdös and Taylor in 1960. This talk will be dedicated to recent progress on this topic. I will in particular discuss the scaling limit of the set of thick points, considerably refining estimates of Dembo, Peres, Rosen and Zeitouni. This scaling limit is described by a random measure which is the analogue of Gaussian multiplicative chaos measures for the local times of planar Brownian motion. I will discuss the construction of this new object and some of its properties. Finally, I will explain a characterisation of this random measure which is a key step in the proof of the above scaling limit.

Janne Junnila (Lausanne): Imaginary chaos. pdf

I will discuss the basic properties of imaginary Gaussian multiplicative chaos, which is a family of random generalized functions that can be viewed as a version of Kahane's Gaussian multiplicative chaos with a purely imaginary intermittency parameter. Such distributions appear for instance in the scaling limit of the spin field of the critical planar XOR-Ising model.

Ellen Powell (Durham): Conformal welding and critical Liouville quantum gravity.

I will discuss a conformal welding problem that relates Gaussian multiplicative chaos measures for the planar Gaussian free field (Liouville quantum gravity) and Schramm--Loewner evolutions. This connection was discovered by Sheffield in the context of subcritical Liouville quantum gravity. I will present a new result with Nina Holden for the critical case, and discuss some consequences.

Avelio Sepúlveda Donoso (Lyon): Imaginary chaos and the level sets of the Gaussian free field.

In this talk, I will discuss the relationship between imaginary Gaussian multiplicative chaos, the level sets of the Gaussian free field and the XOR-Ising model. In particular, I will show how to condition an imaginary chaos on a level sets of the free field and relate this to certain renewal sets of the XOR-Ising model.

Vincent Vargas (Paris): A probabilistic approach of ultraviolet renormalisation in the boundary Sine-Gordon model.

The Sine-Gordon model is a celebrated model of 2d quantum field theory based on the quantization of a cosine interaction term. Though the model has been studied by numerous authors, it is fair to say that a complete mathematical picture is still lacking. In this talk, I will present a novel probabilistic approach to renormalization of the boundary Sine-Gordon model. This new approach, which is based on modern stochastic calculus and a new formula for the cumulants of a random variable (which could be of independent interest), enables to define the correlations of the model in a robust way. Based on a work with H. Lacoin, R. Rhodes.

Christian Webb (Aalto): How much can the eigenvalues of a random Hermitian matrix fluctuate? pdf

One of the most classical results of random matrix theory is Wigner's theorem, which says that the eigenvalues of a large random Hermitian matrix, drawn from the Gaussian Unitary Ensemle (GUE), are approximately distributed according to the semicircle distribution. I will discuss some recent joint work with T. Claeys (UCLouvain) B. Fahs (Imperial College London), and G. Lambert (University of Zürich) describing how much the eigenvalues can fluctuate around the semicircle distribution.

Ofer Zeitouni (Rehovot): Extremes for log-correlated fields: beyond the Gaussian case. pdf

Log-correlated fields appear in the study of characteristic polynomials of random matrices, as well as in problems related to exceptional points for two dimensional Brownian motion and random walks. In these cases, the field is not Gaussian - sometimes a Gaussian field is not far behind while in other cases the field is actually far from Gaussian. I will describe describe some recent work attempting to understand the behavior of the extremes.