Zeros of random linear combinations of OPUC with complex Gaussian coefficients

We study zero distribution of random linear combinations of the form $P_n(z)=\sum_{j=0}^nη_jϕ_j(z),$ in any Jordan region $Ω\subset \mathbb C$. The basis functions $ϕ_j$ are orthogonal polynomials on the unit circle (OPUC) that are real-valued on the real line, and $η_0,\dots,η_n$ 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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