I will discuss results by R. Latala concerning tail behaviour of multivariate polynomials in independent Gaussian variables and show how when combined with classical functional inequalities they give estimates for polynomials and more generally smooth functions with bounded derivatives of higher order for a more general class of non-necessarily product measures. I will also present similar inequalities for polynomials of general sequences of independent subgaussian random variables, which do not possess concentration properties for Lipschitz functions. Based on joint work with P. Wolff (University of Warsaw).

The goals for the series of talks are as follows.

1. To fill in background for and to state the result of Haagerup, Schultz

and Thorbjornsen for polynomials in GUE matrices.

2. To prove the result taking care to explain the most important ``tricks'',

e.g, the linearization trick.

3. To discuss extensions and related results in the literature.

4. To prove some extensions of HST to polynomials in Wigner matrices

(not the best or sharpest possible) using methods worth learning about in

their own right.

5. To point out some open problems in the area.

I will discuss the effect of finite rank perturbations (or "spikes") on the random matrix soft edge, focusing on the phase transition discovered in the complex setting by Baik, Ben Arous and Péché. While small spikes leave the usual Tracy-Widom fluctuations of the top eigenvalues unaltered, large spikes lead to outliers with Gaussian fluctuations. New structure emerges around the transition point with near-critical spikes deforming the soft edge limit. Understanding this transition regime in the real case remained open for some time.

I will describe joint work with B. Virág that treats the phase transition in a general beta setting. With a single spike one obtains the usual limiting random Schrödinger operator on the half-line but with a modified boundary condition depending on the spike; to deal with several spikes we develop a matrix-valued analogue. The resulting deformations of the Tracy-Widom laws can be further characterized in terms of a diffusion (related to Dyson's Brownian motion in the higher-rank case) or a linear parabolic PDE. The latter can be connected with the known Painlevé II structure and, separately, seems effective for numerical evaluation.

Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis concerns large but finite dimensional matrices. Spectacular progress was done in the past two decades to prove universality of random matrices presenting an orthogonal, unitary or symplectic invariance. These models correspond to log-gases with respective inverse temperature 1, 2 or 4. I will report on a joint work with L. Erdős and H.-T. Yau, which yields universality for the log-gases at arbitrary temperature at the microscopic scale. A main step consists in the optimal localization of the particles, and the involved techniques include a multiscale analysis and a local logarithmic Sobolev inequality.

Large dimension expansion of matrix integrals has long been used to study combinatorial objects such as maps, that is graphs sorted by the genus of the surfaces in which they can be properly embedded. In this course, we shall study these expansions. We will first motivate this approach and consider formal expansions. The asymptotics expansions will require a more detailed study of the properties of the spectral measure of random matrices. We shall first consider classical one-matrix models and study the asymptotics of their spectral measure; law of large numbers, central limit theorem, concentration inequalities and large deviations properties. In a second time we shall obtain a full asymptotic expansion of the Cauchy-Stieljes transform of the spectral measure, hence provided a rigorous derivation of the asymptotic topological expansion of matrix integrals in the one matrix case. In the second part of the course, we shall tackle several matrix models, and their application in the enumeration of colored maps and sophisticated combinatorial objects. If time allows, we shall effectively use random matrices to count combinatorial models such as planar maps or loop models.

I give an introduction to quantum diffusion and localization properties of eigenvectors of random band matrices. I give a short overview of the Anderson model, followed by a more in-depth discussion of random band matrices. After a survey of recent results, I sketch the main ideas of the proofs. I focus on two approaches: perturbative renormalization and self-consistent equations for the resolvent combined with fluctuation averaging.

Several settings of the Littlewood-Offord estimate will be discussed in the first half of the talk. These results will then be applied to bound the least singular value for several random matrix models including symmetric matrices and doubly stochastic matrices.

The Harish-Chandra-Itzykson-Zuber integral is a remarkable special function which plays a key role in random matrix theory, where it enters into the description of the spectra of coupled random matrices, Hermitian Wigner matrices, and complex sample covariance matrices. As shown by Guionnet and Zeitouni, the leading asymptotics of the HCIZ integral can be characterized as the solution to a certain variational problem. I will present joint work with I. Goulden and M. Guay-Paquet which relates the HCIZ integral to a classical topic in enumerative geometry, namely counting branched covers of the Riemann sphere with specified branch points and monodromies. In particular, the HCIZ integral can be viewed as a generating function for a desymmetrized version of the double Hurwitz numbers introduced by Okounkov.

In RMT the hard edge refers to the scaling limits of the minimal eigenvalues for matrices of sample covariance type. In the classical invariant ensembles, the limit distributions are characterized by a Bessel kernel and an associated Painleve III equation (as opposed to the better known Airy kernel and Painleve II descriptions at the "soft" edge). We will show that in the general beta setting these descriptions can be replaced by a limiting (random) differential operator and/or the hitting distributions of a related diffusion process. With this in hand various applications of the operator/diffusion picture will be presented, such as sharp tail asymptotics for the beta hard edge laws, the effects of finite rank perturbations to the input matrices, and the link to Painleve III.

Joint work with J. Ramirez and I. Rumanov.

(joint work with A. Guionnet). By solving a free analog of the Monge-Amp`ere equation, we prove a non-commutative analog of Brenier's monotone transport theorem: if an $n$-tuple of self-adjoint non-commutative random variables $Z_{1},...,Z_{n}$ satisfies a regularity condition (its conjugate variables $xi_{1},...,xi_{n}$ should be analytic in $Z_{1},...,Z_{n}$ and $xi_{j}$ should be close to $Z_{j}$ in a certain analytic norm), then there exist invertible non-commutative functions $F_{j}$ of an $n$-tuple of semicircular variables $S_{1},...,S_{n}$, so that $Z_{j}=F_{j}(S_{1},...,S_{n})$. Moreover, $F_{j}$ can be chosen to be monotone, in the sense that $F_{j}=mathscr{D}_{j}g$ and $g$ is a non-commutative function with a positive definite Hessian. In particular, we can deduce that $C^{*}(Z_{1},...,Z_{n})cong C^{*}(S_{1},...,S_{n})$ and $W^{*}(Z_{1},...,Z_{n})cong L(mathbb{F}(n))$. Thus our condition is a useful way to recognize when an $n$-tuple of operators generate a free group factor. We obtain as a consequence that the q-deformed free group factors $Gamma_{q}(mathbb{R}^{n})$ are isomorphic (for sufficiently small $q$, with bound depending on $n$) to free group factors. We also partially prove a conjecture of Voiculescu by showing that free Gibbs states which are small perturbations of a semicircle law generate free group factors. Lastly, we show that entrywise monotone transport maps for certain Gibbs measure on matrices are well-approximated by the matricial transport maps given by free monotone transport.

The prototypical joint density of eigenvalues of a random matrix contains as a factor a power of the absolute value of the Vandermonde determinant in the variables of integration. The most nuanced statistical information can be derived when this power, denoted beta, takes the value 1, 2 or 4. Of particular importance in this talk, is the fact that, when beta is 1 or 4, the marginal densities (vis correlation functions) can be expressed as Pfaffians of antisymmetric matrices formed from a (matrix) kernel. That is, in this situation, the eigenvalues form a Pfaffian point process. In this talk, I will explain one path to the derivation of the Pfaffian point process from the relatively basic fact that the partition functions for these ensembles are themselves expressed as Pfaffians. After introducing the notion of the hyperpfaffian, and showing that, when beta is a square integer, the partition function is a hyperpfaffian, I will ask the question: Does there a hyperpfaffian point process for the eigenvalues when beta is a square integer. I will give evidence for and against such a thing, and sketch out a possible method of proof, assuming the question has an affirmative answer.

We consider several classes of random self-adjoint and unitary operators and investigate their microscopic eigenvalue distribution. We show that some of these operators exhibit a transition in their microscopic eigenvalue distribution, depending on the properties of the corresponding spectral measures. In the case of pure point spectral measures, the microscopic eigenvalue distribution is Poisson (no correlation). As the spectral measures approach an absolutely continuous measure, the repulsion between the eigenvalues increases and the microscopic eigenvalue distribution converges to the clock (or "picket fence") distribution.

Random regular graphs models play an important role in random graph theory and applications, but the models are harder to be investigated than the corresponding Erdos - Renyi models due to the lack of independence between the edges. However the two models share many similar properties. In this talk we will discuss the comparison method, which transform properties of regular graphs into properties of Erdos Renyi ones. Several applications and open problems will be discussed, including our new result on the convergence of empirical spectral distribution of biregular bipartite graphs.

By the Hilbert-Polya conjecture the critical zeros of the Riemann zeta function correspond to the eigenvalues of a self adjoint operator. By a conjecture of Dyson and Montgomery the critical zeros (after a certain rescaling) look like the bulk eigenvalue limit point process of the Gaussian Unitary Ensemble. It is natural to ask if this point process can we described as the spectrum of a random self adjoint operator. I will show that this is indeed the case: for any beta>0 the bulk limit of the Gaussian beta ensemble can be obtained as the spectrum of a self adjoint random differential operator. I will describe the operator and show how to derive it as the limit of operators corresponding to finite ensembles in the circular beta ensemble case.

(Joint with Balint Virag)

We will discuss matrix models for beta ensembles, as well as their infinite limits. The limits are random operators; studying these operators will give access to asymptotic questions about the eigenvalue distribution of finite matrices.

This series will highlight the main steps in the approach to universality based on dynamics and comparisons of Green functions. The emphasis will be on the basic steps, not on sharpest results.