États de la recherche en matrices aléatoires

Institut Henri Poincaré, Paris – December 1-4, 2014 – Amphi Hermite

Welcome to the 2014 edition of the SMF conference series États de la recherche. This 2014 edition is devoted to a state of the art in random matrix theory and some applications such as random growth models and random graph theory. The conference consists in mini-courses for non expert audience including PhD students, and complementary talks.

• The conference will take place in amphitheater Hermite (venue: IHP, Google Maps).
• The registration is free but mandatory (see below).
• A poster is available in PDF.

• Mini-courses
  • Charles Bordenave (CNRS Toulouse, France) - Spectrum of random graphs.
    The spectral properties of large random graphs attracts a growing attention in various branches of mathematics, computer science and statistical physics of disordered systems. The motivation comes from many distinct directions, notably random matrices, random Schrödinger operators, quantum ergodicity or algorithmic spectral graph theory. In this mini-course, we will present some recent results on eigenvalues and eigenvectors of adjacency matrices of finite graphs and conclude with some open problems. (Lecture Notes)
  • Alexei Borodin (MIT, USA) - Macdonald processes.
    Our goal is to explain how certain basic representation theoretic ideas and constructions encapsulated in the form of Macdonald processes lead to nontrivial asymptotic results in various `integrable' probabilistic problems. Examples include dimer models, general beta random matrix ensembles, and various members of the (2+1)d anisotropic KPZ and (1+1)d KPZ universality classes, such as growing stepped surfaces, q-TASEP, q-PushASEP, and directed polymers in random media. Slides 1/3 Slides 2/3 Slides 3/3
  • Antti Knowles (ETHZ, Switzerland) - Local laws in random matrix theory.
    We will give an introduction to local laws in random matrix theory, focusing on the simple case of Wigner matrices. The first part of the course is devoted to the proof of the local law. In the second part, we will give applications to eigenvector delocalization and the universality of the eigenvalue distribution. If time allows, we will discuss further applications to deformed matrix ensembles.
• Complementary talks
  • Paul Bourgade (NYU, USA) - From the mesoscopic to microscopic scale in random matrix theory.
    I will review recent techniques to access microscopic statistics from mesoscopic ones, concerning both eigenvalues and eigenvectors of random matrices. Homogenization and the coupling method play a key role. This complements Antti Knowles' mini course: local laws are very useful a priori estimates. Slides
  • Cédric Boutillier (UPMC, France) - Dimers on Rail Yard Graphs.
    We introduce a general model of dimer coverings of certain plane bipartite graphs, which we call rail yard graphs. Using a transfer matrix approach and the celebrated boson-fermion correspondence, the model can be reformulated as a Schur process. We obtain explicit expressions for the partition function and for the inverse Kasteleyn matrix, which yields all dimer correlation functions. Plane partitions, domino tilings of the Aztec diamond and pyramid partitions arise as particular cases of our model. In the scaling limit, the model shows a similar universal behaviour as the one described by Okounkov and Reshetikhin for plane partitions, in connection with distributions appearing in random matrices. This is based on a joint work with Jérémie Bouttier, Guillaume Chapuy, Sylvie Corteel and Sanjay Ramassamy. Slides
  • Mireille Capitaine (CNRS Toulouse, France) - Spectrum of large deformed Hermitian random matrices and Free Probability Theory.
    We show how free probability theory sheds light on the spectrum of large deformed random matrices, and provides a unified understanding of several phenomena. Slides

Some of the lecture notes were transformed into a book, entitled “Advanced Topics in Random Matrices”, and published in 2018 in the Panoramas et Synthèses book series of the Société Mathématiques de France.

Registration is over.

• Organizers
  • Béatrice de Tilière (Université Pierre et Marie Curie)
  • Djalil Chafaï (Université Paris-Dauphine)
  • Florent Benaych-Georges (Université Paris-Descartes)
  • Sandrine Péché (Université Paris-Diderot)
• Sponsors
  • Société Mathématique de France (SMF)
  • Agence Nationale de la Recherche (ANR)
  • Institut Henri Poincaré (IHP)
  • Institut Universitaire de France (IUF)

The photo of Wigner in front of a blackboard is borrowed from http://www.math.ethz.ch/u/felder/Research/RandomMatrices.
The colorful image of the Ginibre ensemble was created by FBG.