Real roots of random orthogonal polynomials with exponential weights

TL;DR

This paper studies the distribution of real roots of random orthogonal polynomials with exponential weights, establishing universal asymptotic behaviors and limits for root counting measures using advanced probabilistic and orthogonal polynomial techniques.

Contribution

It introduces new methods applying inverse Littlewood-Offord theory to orthogonal polynomials, establishing universality and almost sure limits for root distributions.

Findings

Universal asymptotics for expected number of real roots

Almost sure convergence of root counting measures

Application of inverse Littlewood-Offord theory to orthogonal polynomials

Abstract

We consider random orthonormal polynomials $$ P_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, . . . , $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep_0)$-moments, and $\{p_n\}_{n=0}^{\infty}$ is the system of orthonormal polynomials with respect to a general exponential weight $W$ on the real line. This class of orthogonal polynomials includes the popular Hermite and Freud polynomials. We establish universality for the leading asymptotics of the expected number of real roots of $P_n$, both globally and locally. In addition, we find an almost sure limit of the measures counting all roots of $P_n.$ This is accomplished by introducing new ideas on applications of the inverse Littlewood-Offord theory in the context of the classical three term recurrence relation for orthogonal polynomials to establish anti-concentration properties, and by adapting the universality methods to the weighted random orthogonal polynomials of the form $W P_n.$