Narrow Results

Consider a random matrix of the form , where M_n is a Wigner matrix and D_n is a real deterministic diagonal matrix (D_n is commonly referred to as an external source in the mathematical physics literature). We study the universality of the local eigenvalue statistics of W_n for a general class of Wigner matrices M_n and diagonal matrices D_n. Unlike the setting of many recent results concerning universality, the global semicircle law fails for this model. However, we can still obtain the universal sine kernel formula for the correlation functions. This demonstrates the remarkable phenomenon that local laws are more resilient than global ones. The universality of the correlation functions follows from a four moment theorem, which we prove using a variant of the approach used earlier by Tao and Vu.

The goal of this paper is to rederive the connection between the Painlevé 5 integrable system and the universal eigenvalues correlation functions of double-scaled Hermitian matrix models, through the topological recursion method. More specifically we prove, to all orders, that the WKB asymptotic expansions of the τ-function as well as of determinantal formulas arising from the Painlevé 5 Lax pair are identical to the large N double scaling asymptotic expansions of the partition function and correlation functions of any Hermitian matrix model around a regular point in the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to O(N^-5).

We explore the boundaries of sine kernel universality for the eigenvalues of Gaussian perturbations of large deterministic Hermitian matrices. Equivalently, we study for deterministic initial data the time after which Dyson’s Brownian motion exhibits sine kernel correlations. We explicitly describe this time span in terms of the limiting density and rigidity of the initial points. Our main focus lies on cases where the initial density vanishes at an interior point of the support. We show that the time to reach universality becomes larger if the density vanishes faster or if the initial points show less rigidity.