Poisson statistics for zeros of random orthogonal polynomials on the unit circle

Mihai Stoiciu California Institute of Technology, USA

We consider random paraorthogonal polynomials on the unit circle {Φ_{n}}_{n≥0} defined by random i.i.d. Verblunsky coefficients α_{0}, α_{1}, …, α_{n−2} uniformly distributed in a disk of radius r < 1 and α_{n−1} uniformly distributed on the unit circle. For any n, the zeros of Φ_{n} are n random points on the unit circle.

We prove that for any e^{iθ} ∈ ∂𝔻 the distribution of the zeros of Φ_{n} in intervals of size O(\frac{1}{n}) near e^{iθ} is the same as the distribution of n independent random points uniformly distributed on the unit circle (i.e., Poisson). This means that, for large n, there is no local correlation between the zeros of the considered random paraorthogonal polynomials.