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Mathematics Colloquium

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Date:  Tuesday, January 15, 2013

Title:  Ziwet Lectures: Universality of random matrices and log-gases

Abstract:  1. Universality of random matrices and log-gases. (January 15 at 1360EH) 2. Dyson's Brownian and De Georgi-Nash-Moser theory of parabolic regularity. (January 16 at 1360EH) Abstract: Eugene Wigner’s revolutionary vision predicted that the energy levels of large complex quan- tum systems exhibit a universal behavior: the statistics of energy gaps depend only on the basic symmetry type of the model. These universal statistics show strong correlations in the form of level repulsion and they represent a new paradigm of point processes that are characteristically different from the Poisson statistics of independent points. Simplified models of Wigner’s thesis have recently become mathematically accessible. For mean field models represented by large random matrices with independent entries, the celebrated Wigner-Dyson-Gaudin-Mehta (WDGM) conjecture asserts that the local eigenvalue statistics are universal. For invariant matrix models, the eigenvalue distributions are given by a log-gas with potential $V$ and inverse temperature $\beta = 1, 2, 4$. For $\beta \not \in \{1, 2, 4 \}$, there is no natural random matrix ensemble behind this model, but the analogue of the WDGM conjecture asserts that the local statistics are independent of $V$. In these lectures, we review the recent solution to these conjectures for both invariant and non-invariant ensembles. We will discuss two different notions of universality in the sense of (i) local correlation functions and (ii) gap distributions. We will demonstrate that the local ergodicity of the Dyson Brownian motion is the intrinsic mechanism behind the universality. Furthermore, we will show that the universality of gap distribution requires to prove a Holder regularity of a discrete parabolic equation with random coefficients. For this purpose, we incorporate the ideas of parabolic regularity via a De Giorgi-Nash-Moser approach.


Speaker:  Horng-Tzer  Yau
Institution:  Harvard University

 

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