Random matrices and the six vertex model, Pavel Bleher (IUPUI)
Abstract: I will review the Riemann-Hilbert approach to random matrix models and its applications. The plan of the lectures is as follows: 1. Introduction to random matrix models. The equilibrium measure and the concentration phenomenon. Distribution of eigenvalues as a determinantal point process. 2. Phase transitions and critical phenomena. The Riemann—Hilbert approach to ensembles of random matrices and the universality of the scaling and double scaling limits of the eigenvalue correlation functions in random matrix models. 3. Applications of random matrix models to the six-vertex model.
Point processes, Pablo Ferrari (UBA)
Abstract: Poisson processes. Gibbs measures, existence, uniqueness and perfect simulation. Factor graphs. Ising model. Widom Rowlinson model on Poisson processes. Random cluster model. Harmonic graphs.
Topological expansions, Alice Guionnet (MIT)
Abstract: Topological expansions were first exhbit by t'Hooft and Brezin-Itzykson-Parisi-Zuber for matrix intgerals. It turns out that they can also be derived for many large dimension interacting system, allowing the study of the underlying point process, such as the famous beta-models. During this course we will describe the theory of topologial expansions and their extensions, as well as applications to point processes.
Determinantal point processes, Manjunath Krishnapur (IISC)
Abstract: TBA.
Fekete points, an overview, Joaquim Ortega Cerdà (UB)
Abstract: I will present an overview of the distribution of Fekete points in several contexts. These are points that minimize a certain energy and that appear naturally in problems ranging from electrostatics to approximation theory. We will overview the results in different contexts, from the clasical weighted Fekete points in the plane to more elaborated versions in complex and Riemannian manifolds. We will explore how the conection to determinantal point processes can be enlightening in their study.
Two-scale Gamma convergence for Coulomb gases and weighted Fekete sets, Etienne Sandier (UPEC)
Abstract: In this mini-course we will firstly present recent results on the local behaviour of weighted Fekete sets and on the local law of Coulomb gases. These rely on the computation of the second term in the expansion by Gamma-convergence of the interaction energy of the points. This second term is the energy of the physical system known as a jellium, and we will present some of its properties and if time allows, results obtained in collaboration with Y.Ge on this topic.
Coulomb gases and renormalized energies, Sylvia Serfaty (UPMC)
Abstract: The Coulomb gas is a statistical mechanics system, where particles interact via a Coulomb kernel and are kept together by a confining potential. In dimension 2, such a system at zero temperature (i.e. examining ground states, or energy minimizers) corresponds to "weighted Fekete sets", while for certain specific values of the temperature, the system coincides with the Ginibre ensemble of random matrices, it can also be studied at any temperature. With Etienne Sandier, we derived a renormalized energy characterizing the microscopic pattern arrangements of the points in this setting. With Borodin, we showed how to characterize random point processes through it. Coulomb gases can also be studied in dimensions 3 and higher (dimension 3 being the most physical), and a renormalized energy be derived, this is joint work with Nicolas Rougerie. The course will focus on all these aspects.
Local regimes for beta matrix models, Mariya Shcherbina (ILT)
Abstract: To be added
Random Nodal Portraits, Mikhail Sodin (Tel Aviv University)
Abstract: We describe the progress and challenges of understanding the zero sets of smooth Gaussian random functions of several real variables. The primary examples are various ensembles of Gaussian real-valued polynomials (algebraic or trigonometric) of large degree, and smooth Gaussian functions on the Euclidean space with translation-invariant distribution. This might be thought as a statistical version of Hilbert's 16th problem. The course will be based on joint works with Fedor Nazarov.
The spectra of random sparse graphs, Balint Virag (TU)
Abstract: This short introduction will cover some old and new results. Why is the spectrum of a graph interesting? When does it contain atoms? How continuous is the spectral measure? When are eigenvectors localized / delocalized?
Extremal processes, Ofer Zeitouni (WIS)
Abstract: We discuss the structure of extremal processes in nodels exhibiting long-range correlations, emphasizing the role of decorated Poisson processes.
Abstract: I will review the Riemann-Hilbert approach to random matrix models and its applications. The plan of the lectures is as follows: 1. Introduction to random matrix models. The equilibrium measure and the concentration phenomenon. Distribution of eigenvalues as a determinantal point process. 2. Phase transitions and critical phenomena. The Riemann—Hilbert approach to ensembles of random matrices and the universality of the scaling and double scaling limits of the eigenvalue correlation functions in random matrix models. 3. Applications of random matrix models to the six-vertex model.
Point processes, Pablo Ferrari (UBA)
Abstract: Poisson processes. Gibbs measures, existence, uniqueness and perfect simulation. Factor graphs. Ising model. Widom Rowlinson model on Poisson processes. Random cluster model. Harmonic graphs.
Topological expansions, Alice Guionnet (MIT)
Abstract: Topological expansions were first exhbit by t'Hooft and Brezin-Itzykson-Parisi-Zuber for matrix intgerals. It turns out that they can also be derived for many large dimension interacting system, allowing the study of the underlying point process, such as the famous beta-models. During this course we will describe the theory of topologial expansions and their extensions, as well as applications to point processes.
Determinantal point processes, Manjunath Krishnapur (IISC)
Abstract: TBA.
Fekete points, an overview, Joaquim Ortega Cerdà (UB)
Abstract: I will present an overview of the distribution of Fekete points in several contexts. These are points that minimize a certain energy and that appear naturally in problems ranging from electrostatics to approximation theory. We will overview the results in different contexts, from the clasical weighted Fekete points in the plane to more elaborated versions in complex and Riemannian manifolds. We will explore how the conection to determinantal point processes can be enlightening in their study.
Two-scale Gamma convergence for Coulomb gases and weighted Fekete sets, Etienne Sandier (UPEC)
Abstract: In this mini-course we will firstly present recent results on the local behaviour of weighted Fekete sets and on the local law of Coulomb gases. These rely on the computation of the second term in the expansion by Gamma-convergence of the interaction energy of the points. This second term is the energy of the physical system known as a jellium, and we will present some of its properties and if time allows, results obtained in collaboration with Y.Ge on this topic.
Coulomb gases and renormalized energies, Sylvia Serfaty (UPMC)
Abstract: The Coulomb gas is a statistical mechanics system, where particles interact via a Coulomb kernel and are kept together by a confining potential. In dimension 2, such a system at zero temperature (i.e. examining ground states, or energy minimizers) corresponds to "weighted Fekete sets", while for certain specific values of the temperature, the system coincides with the Ginibre ensemble of random matrices, it can also be studied at any temperature. With Etienne Sandier, we derived a renormalized energy characterizing the microscopic pattern arrangements of the points in this setting. With Borodin, we showed how to characterize random point processes through it. Coulomb gases can also be studied in dimensions 3 and higher (dimension 3 being the most physical), and a renormalized energy be derived, this is joint work with Nicolas Rougerie. The course will focus on all these aspects.
Local regimes for beta matrix models, Mariya Shcherbina (ILT)
Abstract: To be added
Random Nodal Portraits, Mikhail Sodin (Tel Aviv University)
Abstract: We describe the progress and challenges of understanding the zero sets of smooth Gaussian random functions of several real variables. The primary examples are various ensembles of Gaussian real-valued polynomials (algebraic or trigonometric) of large degree, and smooth Gaussian functions on the Euclidean space with translation-invariant distribution. This might be thought as a statistical version of Hilbert's 16th problem. The course will be based on joint works with Fedor Nazarov.
The spectra of random sparse graphs, Balint Virag (TU)
Abstract: This short introduction will cover some old and new results. Why is the spectrum of a graph interesting? When does it contain atoms? How continuous is the spectral measure? When are eigenvectors localized / delocalized?
Extremal processes, Ofer Zeitouni (WIS)
Abstract: We discuss the structure of extremal processes in nodels exhibiting long-range correlations, emphasizing the role of decorated Poisson processes.
The titles of the plenary and invited talks will be provided later on.
Yacin Ameur (Lund University)
Diego Armentano (UDELAR)
Alexander Borichev (Aix-Marseille Université)
Jerry Buckley (King's College London)
Zakhar Kabluchko (UUlm)
José León (UCV)
Alon Nishry (Michigan University)
Ron Peled (Tel Aviv University)
Daniel Remenik (Universidad de Chile)
Igor Wigman (King's College London)
