The courses 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 will have contributed talks from selected young participants.

Furthermore, a final session will run on June 16 to discuss perspectives and open problems.

Click here to download the schedule of the School.

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.

[Slides of the lectures (PDF)]

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 Genève)

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.

[Slides from introductory lecture (PDF), lecture notes (PDF)]

We analyze the distribution of eigenvectors for mesoscopic, mean-field perturbations of diagonal matrices in the bulk of the spectrum. We prove that the eigenvectors entries are asymptotically Gaussian with a specific variance, localizing them onto a small, explicit, part of the spectrum. For a well spread initial spectrum, this variance profile universally follows a heavy-tailed Cauchy distribution. The proof relies on a priori local laws for this model and the eigenvector moment flow.

Florian Dorsch (Friedrich-Alexander-Universität Erlangen-Nürnberg)

A sequence of large invertible matrices given by a small random perturbation around a fixed diagonal and positive matrix induces a random dynamics on a high-dimensional sphere. For a certain class of rotationally invariant random perturbations it is shown that the dynamics approaches the stable fixed points of the unperturbed matrix up to errors even if the strength of the perturbation is large compared to the relative increase of nearby diagonal entries of the unperturbed matrix specifying the local hyperbolicity. This work is motivated by the (long-term) aim of controlling the growth of the finite volume eigenfunctions of the Anderson model in the weak coupling regime of disorder. This is a joint work with Hermann Schulz-Baldes.

Mareike Lager (Institute for Applied Mathematics, Bonn)

We consider a random band matrix ensemble in two and three dimensions, in the limit of infinite volume and fixed but large band width. For this model, we discuss rigorous results on the averaged density of states obtained in [2] and [1]. The main steps of the proof are a supersymmetric dual representation, a saddle point analysis and a suitable cluster expansion. We compare the results and proofs with respect to the dimension. This is a joint work with M. Disertori.

[1] M. Disertori and M. Lager. Density of States for Random Band Matrices in Two Dimensions.

[2] M. Disertori, H. Pinson, and T. Spencer. Density of states for random band matrices.

Axel Saenz Rodriguez (University of Virginia)

This talk is based on a collaboration with Leo Petrov (UVa) and Alisa Knizel (Columbia). We introduce a Directed First Passage Percolation model, a discrete time and space TASEP model, and a Higher-Spin Stochastic Six Vertex with probabilities given by Schur-symmetric functions. We use determinantal formulas to prove the limit shape for the models and find the Tracy-Widom distribution under the proper scaling. We are also able to extend these results (in distribution) to other models outside the scope of Schur processes by applying some recent result from Leo's work.

Vittoria Silvestri (University of Cambridge)

The Hastings-Levitov planar aggregation models describe growing random clusters on the complex plane, built by iterated composition of random conformal maps. A striking feature of these models is that they can be used to define natural off–lattice analogues of several fundamental discrete models, such as the Eden model or Diffusion Limited Aggregation, by tuning the correlation between the defining maps appropriately. In this talk I will discuss shape theorems and fluctuations of large clusters in the weak correlation regime. Based on joint work with James Norris (Cambridge) and Amanda Turner (Lancaster).

Alessio Squarcini (Max-Planck-Institute for Intelligent Systems, Stuttgart)

In this talk we illustrate the exact theory of phase separation for systems of classical statistical mechanics in two dimensions. Low energy properties of two-dimensional field theory are used to determine exact and universal results for order parameter profiles and interfacial fluctuations for the different universality classes in various geometries. As a specific application of our framework we consider near-critical planar systems with boundary conditions inducing phase separation. While order parameter correlations decay exponentially in pure phases, we show by direct field theoretical derivation how phase separation generates long range correlations in the direction parallel to the interface, and determine their exact analytic form. The latter leads to specific contributions to the structure factor of the interface.

Depending on time availability we will provide an overview on other interesting applications of the theory. Specific examples include: systems allowing for single interfaces (as the Ising model), the formation of an intermediate wetting layer, interfacial wetting transitions and interfaces with end-points on a wedge-like boundary. Results available from the lattice solution of the Ising model on the plane and on the half-plane are recovered as a particular case.

Xufan Zhang (Brown University, Rhode Island)

As a generalization of domino shuffling, we construct discrete time global dynamics on periodic dimer models, and prove their hydrodynamic limits.