Variance of real zeros of random orthogonal polynomials for varying and exponential weights

Abstract

We determine the asymptotics for the variance of the number of zeros of random linear combinations of orthogonal polynomials of degree at most n associated with varying weights $\left\{e^{-2n{Q}_n}\right\}$, with Gaussian coefficients. We deduce asymptotics of the variance for fixed exponential weights $e^{-2Q}$. In particular, we show that very generally, the variance is asymptotic to $Cn$, where the constant C involves a universal constant and an equilibrium density associated with the weight(s).