Zeros of orthogonal polynomials

Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)

DORIN GHISA Riemann Zeta function is one of the most studied transcendental functions, having in view its many applications in number theory,algebra, complex analysis, statistics, as well as in physics. Another reason why this function has drawn so much attention is the celebrated Riemann conjecture regarding its non trivial zeros, which resisted proof or disproof until now. There has been lately a lot of progress in the sense of proving its truthfulness.

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The Riemann zeta function

A Dynamical Approach to Random Matrix Theory

Limit Theorems of Probability Theory.

We prove the edge universality of the beta ensembles for any ${\beta \ge 1}$, provided that the limiting spectrum is supported on a single interval, and the external potential is ${\fancyscript{C}^4}$ and regular. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class ${\fancyscript{C}^4}$.

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Edge Universality of Beta Ensembles

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

/pdf/universality-of-general-b-ensembles-31qshzqix2.pdf

Universality of general β-ensembles

We prove that the distribution of eigenvectors of generalized Wigner matrices is universal both in the bulk and at the edge. This includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries. The proof relies on analyzing the eigenvector flow under the Dyson Brownian motion. The key new ideas are: (1) the introduction of the eigenvector moment flow, a multi-particle random walk in a random environment, (2) an effective estimate on the regularity of this flow based on maximum principle and (3) optimal finite speed of propagation holds for the eigenvector moment flow with very high probability.

The Eigenvector Moment Flow and Local Quantum Unique Ergodicity

It was recently conjectured by Fyodorov, Hiary and Keating that the maximum of the characteristic polynomial on the unit circle of a ${N\times N}$ random unitary matrix sampled from the Haar measure grows like ${CN/({\rm log} N)^{3/4}}$ for some random variable C. In this paper, we verify the leading order of this conjecture, that is, we prove that with high probability the maximum lies in the range ${[N^{1 - \varepsilon},N^{1 + \varepsilon}]}$, for arbitrarily small ${\varepsilon}$.

Maximum of the Characteristic Polynomial of Random Unitary Matrices

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

Paul Bourgade

Papers

Edge Universality of Beta Ensembles

Universality of general β-ensembles

The Eigenvector Moment Flow and Local Quantum Unique Ergodicity

Maximum of the Characteristic Polynomial of Random Unitary Matrices

Universality of General $\beta$-Ensembles