We study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of n ×n Hermitian matrices as n → ∞. Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al for orthogonal polynomials with varying weights, we show first that if the support of the Density of States of the model consists of q ≥ 2 intervals, then in the global regime the variance of statistics is a quasiperiodic function of n as n → ∞ generically in the potential, determining the model. We show next that the exponent of the Laplace transform of the probability law is not in general 1/2 × variance, as it should be if the Central Limit Theorem would be valid, and we find the asymptotic form of the Laplace transform of the probability law in certain cases.

We prove the universality of the β-ensembles with convex analytic potentials and for any β> 0, i.e. we show that the spacing distributions of log-gases at any inverse temperature β coincide with those of the Gaussian β-ensembles.

The asymptotic behavior of polynomials that are orthogonal with respect to a slowly decaying weight is very different from the asymptotic behavior of polynomials that are orthogonal with respect to a Freud-type weight. While the latter has been extensively studied, much less is known about the former. Following an earlier investigation into the zero behavior, we study here the asymptotics of the density of states in a unitary ensemble of random matrix with a slowly decaying weight. This measure is also naturally connected with the orthogonal polynomials. It is shown that, after suitable rescaling, the weak limit is the same as the weak limit of the rescaled zeros. 1

We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U- and V-(von Mises) statistics of eigenvalues of random matrices as their size tends to in nity. We show rst that for a certain class of test functions (kernels), determining the statistics, the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics. We then check the conditions of the reduction statements for several most known ensembles of random matrices, The reduction phenomenon is well known in statistics, dealing with i.i.d. random variables. It is of interest that an analogous phenomenon is also the case for random matrices, whose eigenvalues are strongly dependent even if the entries of matrices are independent. 1

We prove the bulk universality of the β-ensembles with non-convex regular analytic potentials for any β> 0. This removes the convexity assumption appeared in the earlier work [6]. The convexity condition enabled us to use the logarithmic Sobolev inequality to estimate events with small probability. The new idea is to introduce a “convexified measure ” so that the local statistics are preserved under this convexification.

We show that near a point where the equilibrium density of eigenvalues of a matrix model behaves like y ∼ xp/q, the correlation functions of a random matrix, are, to leading order in the appropriate scaling, given by determinants of the universal (p, q)-minimal models kernels. Those (p, q) kernels are written in terms of functions solutions of a linear equation of order q, with polynomial coefficients of degree ≤ p. For example, near a regular edge y ∼ x1/2, the (1, 2) kernel is the Airy kernel. Those kernels are associated to the (p, q) minimal model, i.e. the (p, q) reduction of the KP hierarchy solution of the string equation. Here we consider only the 1-matrix model, for which q = 2.

Multivariable generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schrödinger operators for Calogero-Sutherland-type quantum systems. For the generalized Hermite and Laguerre polynomials the multidimensional analogues of many classical results regarding generating functions, differentiation and integration formulas, recurrence relations and summation theorems are obtained. We use this and related theory to evaluate the global limit of the ground state density, obtaining in the Hermite case the Wigner semi-circle law, and to give an explicit solution for an initial value problem in the Hermite and Laguerre case. 1

Abstract: We prove the existence of a 1/N expansion to all orders in β matrix models with a confining, offcritical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the “topological recursion ” derived in Chekhov and Eynard (JHEP 0612:026, 2006). Our method relies on the combination of a priori bounds on the correlators and the study of Schwinger-Dyson equations, thanks to the uses of classical complex analysis techniques. These a priori bounds can be derived following (Boutet de Monvel et al. in J Stat Phys 79(3–4):585–611, 1995; Johansson in Duke Math J 91(1):151–204, 1998; Kriecherbauer and Shcherbina in Fluctuations of eigenvalues of matrix models and their applications, 2010) or for strictly convex potentials by using concentration of measure (Anderson et al. in An introduction to random matrices, Sect. 2.3, Cambridge University Press, Cambridge, 2010). Doing so, we extend the strategy of Guionnet and Maurel-Segala (Ann Probab 35:2160–2212, 2007), from the hermitian models (β = 2) and perturbative potentials, to general β models. The existence of the first correction in 1/N was considered in Johansson (1998) and more recently in Kriecherbauer and Shcherbina (2010). Here, by taking similar hypotheses, we extend the result to all orders in 1/N. 1.