Asymptotic zero distribution of random orthogonal polynomials

TL;DR

This paper studies the asymptotic distribution of zeros of random orthogonal polynomials, establishing conditions under which these zeros converge to the equilibrium measure of a compact set, with optimal results and extensions to multivariable cases.

Contribution

It provides the sharpest conditions for the convergence of zeros of random orthogonal polynomials to the equilibrium measure, including multivariable generalizations and almost sure convergence results.

Findings

Zeros of random orthogonal polynomials converge to equilibrium measure under specific conditions.

Optimal conditions identified for convergence failure when not met.

Extension of results to multivariable polynomials and almost sure convergence.

Abstract

We consider random polynomials of the form $H_n(z)=\sum_{j=0}^n\xi_jq_j(z)$ where the $\{\xi_j\}$ are i.i.d non-degenerate complex random variables, and the $\{q_j(z)\}$ are orthonormal polynomials with respect to a compactly supported measure $\tau$ satisfying the Bernstein-Markov property on a regular compact set $K \subset \mathbb{C}$. We show that if $\mathbb{P}(|\xi_0|>e^{|z|})=o(|z|^{-1})$, then the normalized counting measure of the zeros of $H_n$ converges weakly in probability to the equilibrium measure of $K.$ This is the best possible result, in the sense that the roots of $G_n(z)=\sum_{j=0}^n\xi_jz^j$ fail to converge in probability to the appropriate equilibrium measure when the above condition on the $\xi_j$ is not satisfied. In addition, we give a multivariable version of this result. We also consider random polynomials of the form $\sum_{k=0}^n\xi_kf_{n,k}z^k$, where the coefficients $f_{n,k}$ are complex constants satisfying certain conditions, and the random variables $\{\xi_k\}$ satisfy $\mathbb{E} \log(1 + |\xi_0|) < \infty$. In this case, we establish almost sure convergence of the normalized counting measure of the zeros to an appropriate limiting measure. Again, this is the best possible result in the same sense as above.