ASYMPTOTIC ZERO DISTRIBUTION OF RANDOM ORTHOGONAL POLYNOMIALS

• 成果类型: Article
• 署名作者: Bloom, Thomas; Dauvergne, Duncan
• 署名单位: University of Toronto
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1337
• 发表日期: 2019
• 页码: 3202-3230
• 关键词: equidistribution

摘要

We consider random polynomials of the form H-n(z) = Sigma(n)(j=0)xi(j)q(j)(z) where the {xi(j)} are i.i.d. nondegenerate complex random variables, and the {q(j)(z)} are orthonormal polynomials with respect to a compactly supported measure tau satisfying the Bernstein-Markov property on a regular compact set K subset of C. We show that if P(|xi(0)| > e(vertical bar z vertical bar)) = o(vertical bar z vertical bar(-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) = Sigma(n)(j=0)xi(j)z(j) fail to converge in probability to the appropriate equilibrium measure when the above condition on the xi(j) is not satisfied. We also consider random polynomials of the form Sigma(n xi)(k=0)(k)f(n),(k)z(k),where the coefficients f(n,k) are complex constants satisfying certain conditions, and the random variables {xi(k)} satisfy E log(1+vertical bar xi(0 vertical bar)) < infinity. 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.