Central Limit Theorem for the number of real roots of random orthogonal polynomials

We study the number of real roots of a wide class of random linear combinations of orthogonal polynomials with Gaussian coefficients. The orthogonal polynomials in our model are defined by a deterministic measure with compact support on the real line. Using the method of Wiener Chaos, we show that the fluctuation for the number of real roots in the bulk is asymptotically Gaussian, by proving that this number of roots in the intervals inside the support of the orthogonality measure obeys the standard Central Limit Theorem. Wiener Chaos expansions were previously used to prove the CLT for classical ensembles of random trigonometric polynomials, and that approach is generalized in our paper via careful analysis of the correlations by using asymptotics for the reproducing kernels of orthogonal polynomials. A new interesting feature found on this path is that the local correlations for the number of real roots of our random orthogonal polynomials are different. In fact, our local correlations depend on the potential theoretic equilibrium measure for the support of the orthogonality measure.