Real roots of random orthogonal polynomials with exponential weights

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 $W{P}_n.$