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We present some applications of central limit theorems on mesoscopic scales for random matrices. When combined with the recent theory of homogenization for Dyson Brownian Motion, this yields the universality of quantities which depend on the behavior of single eigenvalues of Wigner matrices and $beta$-ensembles. Among the results we obtain are the Gaussian fluctuations of single eigenvalues for Wigner matrices (without an assumption of 4 matching moments) and classical $beta$-ensembles ($beta=1, 2, 4$), Gaussian fluctuations of the eigenvalue counting function, and an asymptotic expansion up to order $o(N^{-1})$ for the expected value of eigenvalues in the bulk of the spectrum. The latter result solves a conjecture of Tao and Vu.
This article begins with a brief review of random matrix theory, followed by a discussion of how the large-$N$ limit of random matrix models can be realized using operator algebras. I then explain the notion of Brown measure, which play the role of t
In this chapter of the Oxford Handbook of Random Matrix Theory we introduce chiral Random Matrix Theories with the global symmetries of QCD. In the microscopic domain, these theories reproduce the mass and chemical potential dependence of QCD. The ma
Let $T$ be a random ergodic pseudometric over $mathbb R^d$. This setting generalizes the classical emph{first passage percolation} (FPP) over $mathbb Z^d$. We provide simple conditions on $T$, the decay of instant one-arms and exponential quasi-indep
We prove infinite-dimensional second order Poincare inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian fields, Steins method and Malliavin calculus. We provide two applications: (i) to a ne
We introduce a simple yet powerful calculational tool useful in calculating averages of ratios and products of characteristic polynomials. The method is based on Dyson Brownian motion and Grassmann integration formula for determinants. It is intended