Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle

We prove that for any $n\times n$ matrix, $A$, and $z$ with $|z|\geq \|A\|$, we have that \|(z-A)^{-1}\|\leq\cot (\frac{\pi}{4n}) \dist (z, \spec(A))^{-1}. We apply this result to the study of random orthogonal polynomials on the unit circle.Comment: 27 page

Fine Structure of the Zeros of Orthogonal Polynomials: A Review

We review recent work on zeros of orthogonal polynomials

How many zeros of a random polynomial are real?

We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve $(1,t,\ldots,t^n)$ projected onto the surface of the unit sphere, divided by $\pi$. The probability density of the real zeros is proportional to how fast this curve is traced out. We then relax Kac's assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the Fubini-Study metric.Comment: 37 page

Universality for mathematical and physical systems

All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this paper we describe some recent history of universality ideas in physics starting with Wigner's model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting.Comment: New version contains some additional explication of the problems considered in the text and additional reference

Average Characteristic Polynomials of Determinantal Point Processes

We investigate the average characteristic polynomial $\mathbb E\big[\prod_{i=1}^N(z-x_i)\big]$ where the $x_i$'s are real random variables which form a determinantal point process associated to a bounded projection operator. For a subclass of point processes, which contains Orthogonal Polynomial Ensembles and Multiple Orthogonal Polynomial Ensembles, we provide a sufficient condition for its limiting zero distribution to match with the limiting distribution of the random variables, almost surely, as $N$ goes to infinity. Moreover, such a condition turns out to be sufficient to strengthen the mean convergence to the almost sure one for the moments of the empirical measure associated to the determinantal point process, a fact of independent interest. As an application, we obtain from a theorem of Kuijlaars and Van Assche a unified way to describe the almost sure convergence for classical Orthogonal Polynomial Ensembles. As another application, we obtain from Voiculescu's theorems the limiting zero distribution for multiple Hermite and multiple Laguerre polynomials, expressed in terms of free convolutions of classical distributions with atomic measures.Comment: 26 page

Poisson brackets of orthogonal polynomials

For the standard symplectic forms on Jacobi and CMV matrices, we compute Poisson brackets of OPRL and OPUC, and relate these to other basic Poisson brackets and to Jacobians of basic changes of variable