Zeros of random linear combinations of OPUC with complex Gaussian coefficients

We study zero distribution of random linear combinations of the form $$P_n(z)=\sum _{j=0}^n{\eta }_j{\varphi }_j(z),$$ in any Jordan region $\Omega \subset \mathbb C$\mathrm{.}Thebasisfunctions$\phi_j$ are orthogonal polynomials on the unit circle (OPUC) that are real-valued on the real line, and ${\eta }_0,\dots ,{\eta }_n$ are complex-valued iid Gaussian random variables. We derive an explicit intensity function for the number of zeros of $P_n$ in $\mathrm{\Omega }$ 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 Szeg\H{o} weights.