Schedule, Notes/Handouts & Videos

Random Matrix Theory and Its Applications I September 13, 2010 - September 17, 2010

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Sep 13, 2010
Monday

09:25 AM - 09:40 AM		Welcome
Location -- Video -- Abstract -- Supplements -- Show Detail
09:40 AM - 10:20 AM		Perturbed Hankel Determinants: Applications to the Information Theory of MIMO Wireless Communications Yang Chen
Location -- Video -- Abstract -- Supplements -- Show Detail
10:20 AM - 10:50 AM		Tea
Location -- Video -- Abstract -- Supplements -- Show Detail
10:50 AM - 11:30 AM		Geodesic distance in planar maps: from matrix models to trees Philippe Di Francesco (University of Illinois at Urbana-Champaign)
Location -- Video -- Abstract Matrix models have proved to be a successful tool for enumerating maps of arbitrary genus, perused by physicists in the context of two-dimensional quantum gravity. We show how planar map results may be rephrased into tree combinatorics, and how they can be generalized to study refined properties such as geodesic distances within maps. Remarkably, the problem is still integrable (and exactly solvable), and shows striking similarities with the orthogonal polynomial solution of the one-matrix model. Supplements -- Show Detail
11:40 AM - 12:20 PM		Exact results in the Random Matrix Theory approach to the theory of chaotic cavities Francesco Mezzadri (University of Bristol)
Location -- Video Abstract Random Matrix Theory has been used extensively to compute physical quantities related to charge fluctuations in the scattering of a particle through a chaotic cavity. In this talk we will present some results concerning the conductance, shot notice and the Wigner time delay. Our computations involve all the three symmetry classes (beta=1,2 and 4) and are exact for arbitrary number of open channels (finite matrix dimension). Supplements -- Show Detail
12:20 PM - 02:15 PM		Lunch
Location -- Video -- Abstract -- Supplements -- Show Detail
02:15 PM - 02:55 PM		Genetics and large random matrices Nicholas Patterson
Location -- Video Abstract We explain why random matrix theory is relevant to genetics, show some examples on real data, and discuss some problems both statistical and related to numerical technique. Supplements -- Show Detail
02:55 PM - 03:45 PM		Tea
Location -- Video -- Abstract -- Supplements -- Show Detail
04:10 PM - 05:10 PM		Beyond the Gaussian Universality Class <b>(at UC Berkeley-60 Evans Hall)</b> Ivan Corwin (Columbia University)
Location -- Video -- Abstract The Gaussian central limit theorem says that for a wide class of stochastic systems, the bell curve (Gaussian distribution) describes the statistics for random fluctuations of important observables. In this talk I will look beyond this class of systems to a collection of probabilistic models which include random growth models, polymers, particle systems, PDEs and matrices, as well as certain asymptotic problems in combinatorics and representation theory. I will explain in what ways these different examples all fall into a single new universality class with a much richer mathematical structure than that of the Gaussian. The full scope and structure of this new universality class is just one of many important open questions being studied at MSRI this fall. Supplements -- Show Detail

Sep 14, 2010
Tuesday

09:30 AM - 10:10 AM

Planar algebras and the Potts model on random graphs
Alice Guionnet (École Normale Supérieure de Lyon)

10:10 AM - 10:40 AM

Tea

10:40 AM - 11:20 AM

Limiting distributions for TASEP, Last Passage Percolation and a few words on universality in KPZ
Sandrine Peche (Université de Paris VII (Denis Diderot))

11:30 AM - 12:10 PM

Extreme gaps in the spectrum of Random Matrices
Gérard Ben Arous (New York University, Courant Institute)

12:10 PM - 02:00 PM

Lunch

02:00 PM - 02:40 PM

Cluster Expansions, Caustics and Counting Graphs
Nicholas Ercolani (University of Arizona)

02:50 PM - 03:30 PM

Orthogonal and symplectic matrix models: universality and other properties
Mariya Shcherbina (B. Verkin Institute for Low Temperature Physics)

03:30 PM - 04:00 PM

Tea

04:00 PM - 04:40 PM

Beta ensembles on the line, edge universality
Brian Rider (Temple University)

04:40 PM - 05:50 PM

Reception

Sep 15, 2010
Wednesday

09:30 AM - 10:10 AM		Determinant expansions for perturbations of finite Toeplitz matrices Estelle Basor (AIM - American Institute of Mathematics)
Location -- Video -- Abstract This talk will describe analogues of Szegö limit theorems for perturbations of Toeplitz matrices. These include Toeplitz plus Hankel matrices. Supplements -- Show Detail
10:10 AM - 10:40 AM		Tea
Location -- Video -- Abstract -- Supplements -- Show Detail
10:40 AM - 11:20 AM		Aspects of Toeplitz and Hankel determinants Igor Krasovsky (Imperial College, London)
Location -- Video Abstract We review the asymptotic behavior of a class of Toeplitz as well as related Hankel and Toeplitz + Hankel determinants which arise in integrable models and other contexts. We discuss Szego, Fisher-Hartwig asymptotics, and a transition between them. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling limits, to Fredholm determinants which appear in the theory of group representations, in random matrices, random permutations and partitions. The connection to Toeplitz determinants helps to evaluate the asymptotics of related Fredholm determinants in situations of interest, and we mention results on the sine, Bessel, and confluent hypergeometric kernel determinants. The talk is based on the joint works with Tom Claeys, Percy Deift, Alexander Its, and Julia Vasilevska. Supplements -- Show Detail
11:30 AM - 12:10 PM		Universality in Non-Hermitian RMT Gernot Akemann (Universität Bielefeld)
Location -- Video Abstract -- Supplements -- Show Detail

Sep 16, 2010
Thursday

09:30 AM - 10:10 AM		Universality of Wigner random matrices via the four moment theorem Terence Tao (University of California, Los Angeles)
Location -- Video Abstract There has been much recent progress on understanding the fine-scale structure of the spectrum of Wigner random matrices (Hermitian random matrices whose upper-triangular entries are independent), including the important localisation results of Erdos-Schlein-Yau, and the universality results that have been obtained both through the local relaxation flow techniques of Erdos-Schlein-Yau, and through the four moment theorem of Van Vu and the speaker. In this talk we will focus on the four moment theorem, which roughly speaking asserts that the local behaviour of two random matrix ensembles are asymptotically equivalent if the entries have moments matching to fourth order. Interestingly, the need for four matching moments is necessary in some senses, but not in others, as we will describe in the talk. Supplements -- Show Detail
10:10 AM - 10:40 AM		Tea
Location -- Video -- Abstract -- Supplements -- Show Detail
10:40 AM - 11:20 AM		Universality of Random Matrices, Dyson Brownian Motion and Local Semicircle Law Horng-Tzer Yau (Harvard University)
Location -- Video Abstract -- Supplements -- Show Detail
11:30 AM - 12:10 PM		(Random) Tri-Diagonal, Doubly Stochastic Matrices, Orthogonal Polynomials and Alternating Permutations Persi Diaconis (Stanford University)
Location -- Video -- Abstract The set of tri-diagonal, doubly stochastic matrices is a compact convex set. Thus, it makes sense to "pick such a matrix uniformly" and ask about its properties (spectral gap, mixing times, minimum entry, ...). This is intimately connected with the combinatorics of alternating matrices. Jacoby polynomials make a serious appearance. All of this is joint work with Philip Matchett Wood. Supplements -- Show Detail
12:10 PM - 02:30 PM		Lunch
Location -- Video -- Abstract -- Supplements -- Show Detail
02:30 PM - 03:10 PM		Maximal eigenvalue in beta ensembles : large deviations and left tail of Tracy-Widom laws Gaëtan BOROT
Location -- Video Abstract The study of loop equations (also called Pastur equations or Schwinger-Dyson equations) is one (of the) powerful technique(s) to determine all order asymptotics of "matrix models", for large size N of the matrices. J.Ambjorn, L.Chekhov, C.Kristjansen, Yu.Makeenko, G.Akemann contributed to the development of this technique. For hermitean matrix models, the general solution of loop equations was found by B. Eynard in 2004-2005 in terms of algebraic geometry of a certain plane curve, related to the density of eigenvalues at large N. Though this construction does not use orthogonal polynomials or integrability, it is deeply related with those structures. Then, it was extended in 2006 by L.Chekhov and B.Eynard to the beta ensembles. In this talk, I shall illustrate this method (called topological recursion) on the simple case of beta ensembles on the real line with a hard edge a, in the one-cut regime. Actually, the partition function is the probability that the maximum eigenvalue in beta ensembles is smaller that a. Its leading behavior at large N is the large deviation function for the maximum eigenvalue, and the subleading orders give corrections to the large deviations. In a joint work with B.Eynard, S.Majumdar and C.Nadal (math-ph/1009.1945), we have obtained explicit expressions for the "dominant terms" of the large deviation function, and probed them numerically. I will also discuss the connection to the left tail of Tracy-Widom laws. This leads to recover known results for the cases of GOE/GUE/GSE, to obtain some new results for general beta, and to discuss universality. Supplements -- Show Detail
03:10 PM - 04:00 PM		Tea
Location -- Video -- Abstract -- Supplements -- Show Detail
04:00 PM - 04:40 PM		Painleve Equations - Nonlinear Special Functions Peter Clarkson
Location -- Video -- Abstract In this talk I shall discuss various properties of the Painleve equations, which can be thought of as nonlinear special functions, in particular those properties which are relevant to Random matrices. Supplements -- Show Detail

Sep 17, 2010
Friday

09:30 AM - 10:10 AM		TBD Pierre van Moerbeke (Brandeis University; Université Catholique de Louvain)
Location -- Video -- Abstract -- Supplements -- Show Detail
10:10 AM - 10:40 AM		Tea
Location -- Video -- Abstract -- Supplements -- Show Detail
10:40 AM - 11:20 AM		Vector equilibrium problem for the two-matrix model Arno Kuijlaars (Katholieke Universiteit Leuven)
Location -- Video Abstract I discuss an equilibrium problem for a vector of three measures that is associated with the two matrix model with a quartic potential. The first component of the minimizer describes the limiting mean eigenvalue density of one of the random matrices in the two matrix model. There is an external field acting on the first measure, an upper constraint on the second measure, and, in the case of a double well quartic potential, an external field on the third measure as well. The latter fact allows for new critical phenomena that are not observed in the usual one matrix model. This is joint work with Maurice Duits (CalTech) and Man Yue Mo (Bristol). Supplements -- Show Detail
11:30 AM - 12:10 PM		Universality Behviour of Solutions of Hamiltonian PDEs in Critical Regimes Tamara Grava (University of Bristol; International School for Advanced Studies (SISSA/ISAS))
Location -- Video Abstract We study the solution of a class of Hamiltonian PDE near critical points and we show that the solution locally does not depend on the initial data and it is described by particular solutions of Painleve equations. Supplements -- Show Detail
12:10 PM - 02:30 PM		Lunch
Location -- Video -- Abstract -- Supplements -- Show Detail
02:30 PM - 03:10 PM		Asymptotics for the Korteweg-de Vries equation and perturbations using Riemann-Hilbert methods Tom Claeys (Université Catholique de Louvain)
Location -- Video Abstract Small dispersion asymptotics for solutions to the Korteweg-de Vries equation can be obtained using Riemann-Hilbert problems. In critical regimes, this leads to asymptotic expansions in terms of Painlev\'e transcendents. I will give an overview of this procedure, and I will discuss some obstacles that occur when considering non-integrable perturbations of the Korteweg-de Vries equation. This is based on joint work with Tamara Grava, and on joint work in progress with Tamara Grava and Ken McLaughlin. Supplements -- Show Detail
03:10 PM - 04:00 PM		Tea
Location -- Video -- Abstract -- Supplements -- Show Detail
04:00 PM - 04:40 PM		Six-vertex model of statistical mechanics and random matrix models Pavel Bleher (Indiana University--Purdue University)
Location -- Video Abstract We will review various results on the exact solution of the six-vertex model with domain wall boundary conditions, obtained with the help of ensembles of random matrices and the Riemann-Hilbert approach. Supplements -- Show Detail

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