Asymptotic zero distribution of random orthogonal polynomials

Abstract

We consider random polynomials of the form H n (z) = n j=0 ξ j q j (z) where the {ξ j } are i.i.d non-degenerate complex random variables, and the {q j (z)} are orthonormal polynomials with respect to a compactly supported measure τ satisfying the Bernstein-Markov property on a regular compact set K ⊂ C. We show that if P(|ξ 0 | > e |z| ) = o(|z| −1 ), then the normalized counting measure of the zeros of H n converges weakly in probability to the equilibrium measure of K. This is the best possible result, in the sense that the roots of G n (z) = n j=0 ξ j z j fail to converge in probability to the appropriate equilibrium measure when the above condition on the ξ j is not satisfied. In addition, we give a multivariable version of this result.We also consider random polynomials of the form n k=0 ξ k f n,k z k , where the coefficients f n,k are complex constants satisfying certain conditions, and the random variables {ξ k } satisfy E log(1 + |ξ 0 |) < ∞. In this case, we establish almost sure convergence of the normalized counting measure of the zeros to an appropriate limiting measure. Again, this is the best possible result in the same sense as above.

An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC

“…It was shown by Wilkins [25], after some intermediate results cited in [25], that there exist constants A p , p ≥ 0, such that E n (R) has an asymptotic expansion of the form Many subsequent results on random polynomials are concerned with relaxing the conditions on random coefficients, see, for example, [13,18,10], or the behavior of the counting measures of zeros of random polynomials as in [21,6,14,5,19,2,20,17,4,9]. Our primary interest lies in studying the expected number of real zeros when the basis is a family of orthogonal polynomials in the spirit of [7,8,26,16].…”

A necessary and sufficient condition for convergence of the zeros of random polynomials

“…As discussed above, the sufficiency of condition (6) for Theorem 1.2 was proven as Theorem 5.3 in [6] for Bernstein-Markov measures τ with regular support K. These proofs can be extended to include all cases of Theorem 1.2 with a few modifications. The proof ideas can also be used to show the convergence in probability in Theorem 1.4 when the sequence {p n } has an additional assumption about root concentration or speed of convergence (i.e.…”

“…Bloom and Dauvergne [6] extended this result by showing that µ Gn converges to µ K in probability for any sequence of polynomials {p j } generated from a measure τ satisfying the Bernstein-Markov property (we will discuss this property later in the introduction) under the condition (3) P( ξ 0 > e n ) = o(n −1 ).…”

“…However, the method of [6] does not extend as easily to asymptotically minimal polynomials without any condition on root concentration or speed of convergence, and does not extend at all to the case of almost sure convergence.…”

Mass Equidistribution for Random Polynomials

“…Random polynomials with non-Gaussian coefficients were also considered by various authors (see eg. [DS,BL,Bay1,Bay3,BD] among others). In [Bay4] for radially symmetric weight functions, we provided a necessary and sufficient condition on random coefficients for equilibrium distribution of zero divisors of random polynomials (see also [BCM] for the line bundle setting).…”