The school will start on the morning of June 11 and will finish on June 15 in the early afternoon. Each course will consist of four lectures (6 hours in total).

In addition to the main courses, we plan to have contributed talks from selected young participants.

In the 1950's Eugene Wigner made a fundamental observation that the local eigenvalue statistics of sufficiently complex quantum systems exhibit a new type of universality. This celebrated Wigner-Dyson-Mehta statistics has been identified for Gaussian mean field random matrix ensembles in the 1960's by explicit calculations. Going beyond the Gaussian regime required to develop several new techniques that have eventually led to the proof of the Wigner-Dyson-Mehta conjecture on the universality of the local eigenvalue statistics of Wigner matrices. In recent years, Wigner matrices have been studied in increasing generality by gradually relaxing the original conditions that required independent, identically distributed entries. We analyze the key equation, the so-called matrix Dyson equation, that governs the density of states and the behavior of resolvent matrix elements of the corresponding ensemble. As an application, we present local laws and local spectral universality for random matrices with correlated entries.

Ron Peled (Tel Aviv University)

We consider lattice random surface models with nearest-neighbor interactions depending on the gradient of the surface (sometimes called Ginzburg-Landau interface models). Such surfaces form the simplest example of statistical mechanics models with non-compact spin space, and their study is further motivated by their use as effective models for the interfaces separating different phases at thermal equilibrium. The case of quadratic interaction potential leads to the lattice Gaussian free field whose theory is highly developed. Our main object of study in the course is the universal behavior of random surfaces under general interaction potentials. We present classical as well as recent results on questions of localization vs. delocalization, the maximum of the surface, its scaling limit and the roughening transition occurring for integer-valued surfaces in two dimensions.

Fabio Toninelli (Université Lyon 1)

This series of lectures will focus on stochastic (Markovian) reversible dynamics of random interfaces. In statistical physics, such stochastic processes model the evolution of boundaries between coexisting thermodynamic phases (e.g. domain walls in the Ising model). After a heuristic introduction to the general picture (mixing time, hydrodynamic limit, fluctuations, large deviations...), we will discuss some examples that can be treated mathematically. Specifically, we will concentrate mostly on the dynamics of some discrete, two-dimensional interface models that are tightly related to fully-packed dimer models on two-dimensional lattices.

Yvan Velenik (Université de Genčve)

The Ornstein-Zernike theory provides a nonperturbative framework to rigorously analyze many central objects in equilibrium statistical mechanics (interfaces in 2d spin systems, spin correlations, polymers, etc.). The lecture will start with a derivation of the Ornstein-Zernike theory in a simple case (probably the self-avoiding walk, or Bernoulli percolation), and the explanation of the necessary adjustments in more complex situations. I'll then explain in some detail several recent applications of this theory to the Ising and Potts models.