%ALubinsky, Doron%APritsker, Igor%AXie, Xiaoyou%BJournal Name: Mathematical proceedings of the Cambridge Philosophical Society; Journal Volume: 164
%D2018%I
%JJournal Name: Mathematical proceedings of the Cambridge Philosophical Society; Journal Volume: 164
%K
%MOSTI ID: 10092094
%PMedium: X
%TExpected Number of Real Zeros for Random Orthogonal Polynomials
%XWe study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with 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 is given by the orthonormal polynomials associated with a compactly supported Borel measure on the real line, or associated with a Freud weight, then random linear combinations have $n/\sqrt{3} + o(n)$ expected real zeros. We prove that the same asymptotic relation holds for all random orthogonal polynomials on the real line associated with a large class of weights, and give local results on the expected number of real zeros. We also show that the counting measures of properly scaled zeros of these random polynomials converge weakly to either the Ullman distribution or the arcsine distribution.
%0Journal Article