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We study the expected number of real zeros for random linear combinations oforthogonal polynomials. It is well known that Kac polynomials, spanned bymonomials 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 basisis given by orthonormal polynomials associated to a finite Borel measure withcompact support on the real line, then random linear combinations have$n/\sqrt{3} + o(n)$ expected real zeros under mild conditions. We prove thatthe latter asymptotic relation holds for all random orthogonal polynomials onthe real line associated with Freud weights, and give local results on theexpected number of real zeros. We also show that the counting measures ofproperly scaled zeros of random Freud polynomials converge weakly to the Ullmandistribution.

Expected number of real zeros for random Freud orthogonal polynomials