Expected number of real zeros for random Freud orthogonal polynomials

TL;DR

This paper establishes the asymptotic behavior of the expected number of real zeros for random Freud orthogonal polynomials, showing they follow a linear relation and converge to the Ullman distribution.

Contribution

It proves that Freud orthogonal polynomials' random linear combinations have an expected zero count proportional to the degree, extending known results to Freud weights.

Findings

Expected zeros grow linearly with degree for Freud polynomials.

Zeros' counting measures converge to the Ullman distribution.

Local results on the distribution of zeros are provided.

Abstract

We 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 $n$. On the other hand, if the basis is given by orthonormal polynomials associated to a finite Borel measure with compact support on the real line, then random linear combinations have $n/\sqrt{3} + o(n)$ expected real zeros under mild conditions. We prove that the latter asymptotic relation holds for all random orthogonal polynomials on the real line associated with Freud weights, and give local results on the expected number of real zeros. We also show that the counting measures of properly scaled zeros of random Freud polynomials converge weakly to the Ullman distribution.