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/π + o(1)) log n expected real zeros in terms of the degree n. On the other hand, if the basis is given by Legendre (or more generally by Jacobi) polynomials, then random linear combinations have n/ √ 3 + o(n) expected real zeros. We prove that the latter asymptotic relation holds universally for a large class of random orthogonal polynomials on the real line, and also give more general local results on the expected number of real zeros.

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/π+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 defined on the whole real line, then random linear combinations have n/ √ 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.

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/π + 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/ √ 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.

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