UPDATE (June 18th): links to notes and slides for the
lectures by L. Erdős and Y. Velenik have been added below.
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.
Abstracts of the main courses
László
Erdős (IST Austria)
Spectral analysis of general random matrices via the matrix Dyson
equation
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)
Fluctuations of random surfaces
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)
Large-scale dynamics of random interfaces
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)
Ornstein-Zernike theory and some applications
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)]
Abstracts of the contributed talks
Lucas Benigni (LPSM, Université Paris-Diderot)
Eigenvectors distribution and quantum unique ergodicity for
deformed Wigner matrices
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)
Random perturbations of hyperbolic dynamics
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)
Random Band Matrices and Supersymmetry
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. Ann. Henri Poincaré 18(7):2367–2413,
2017.
[2] M. Disertori, H. Pinson, and T. Spencer. Density of states for
random band matrices. Comm. Math. Phys. 232(1):83–124,
2002.
Axel
Saenz Rodriguez (University of Virginia)
Geometric/Bernoulli Growth Process from Schur-Processes
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)
Recent progress on Laplacian growth models
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)
Phase separation and interface structure in two dimensions -
Exact results from field theory
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)
Hydrodynamic limits of discrete time dimer dynamics
As a generalization of domino shuffling, we construct discrete time
global dynamics on periodic dimer models, and prove their
hydrodynamic limits.