Zeros of random linear combinations of OPUC with complex Gaussian coefficients

2016/08/09 by Yeager, Aaron M.
#Complex Variables (math.CV) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1608.02805

Abstract

We study zero distribution of random linear combinations of the form P_n(z)=∑_j=0ⁿη_jϕ_j(z), in any Jordan region Ω⊂ \mathbb C. The basis functions ϕ_j are orthogonal polynomials on the unit circle (OPUC) that are real-valued on the real line, and η₀,…,η_n are complex-valued iid Gaussian random variables. We derive an explicit intensity function for the number of zeros of P_n in Ω for each fixed n. Using the Christoffel-Darboux formula, the intensity function takes a very simple shape. Moreover, we give the limiting value of the intensity function when the orthogonal polynomials are associated to Szegő weights.

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