Real roots of random orthogonal polynomials with exponential weights

Abstract

We consider random orthonormal polynomials

Pₙ(x) = ₙ∑ᵢ=₀ ξᵢpᵢ(x),

where ξ₀, . . . , ξ₀ are independent random variables with zero mean, unit variance and uniformly bounded (2+ε₀)-moments, and {pn}∞ₙ=₀ 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 Pₙ, both globally and locally. In addition, we find an almost sure limit of the measures counting all roots of Pₙ. 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 WPₙ.