Statistics of the number of real zeros of random orthogonal polynomials

Abstract

We study the expected number of real zeros for random linear combinations of orthogonal polynomials Pn(x)=sum_{j=0}^n cjpj(x), where {pj(x)}{j=0}^\infinity is a set of orthonormal polynomials with respect to some measure, supported on the real line, and {cj}{j=0}^\infinity is a set of i.i.d. (independently identically distributed) random variables. It is well known that Kac polynomials Pn(x)=sum_{j=0}^n cjx^j, where {cj}{j=0}^\infinity is a set of i.i.d. Gaussian coefficients, have only (2/pi + o(1))log n expected real zeros in terms of the degree n. If the basis {pj(x)}{j=0}^\infinity is given by the orthonormal polynomials associated with a compactly supported Borel measure mu on the real line or associated with a Freud weight defined on the whole real line, then random linear combinations have n/sqrt{3} + o(n) expected real zeros. We also prove that the same asymptotic relation holds for all random orthogonal polynomials on the real line associated with a large class of exponential weights. It reveals the universality of the expected number of real zeros for random orthogonal polynomials. On the other hand, we give local results on the expected number of real zeros in all considered cases and show that the normalized counting measures of (properly scaled) real zeros of these random polynomials converge weakly.