Zeros of random linear combinations of entire functions with complex Gaussian coefficients

Abstract

We study zero distribution of random linear combinations of the form $$P_n(z)=\sum _{j=0}^n{\eta }_j{f}_j(z),$$ in any Jordan region $\Omega \subset \mathbb{C}$. The basis functions $f_j$ are entire functions 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 $\Omega$ for each fixed $n$. Our main application is to polynomials orthogonal on the real line. 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.

Cited by in corpus (1)

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