Zeros of random analytic functions

Abstract

[en] In this project we deal with random analytic functions. Here we specifically use Gaussian analytic functions. Without technicalities, a GAF $f$ (for short) is a random holomorphic function on a region of $\mathbb{C}$ such that $( f ( z 1 ) , ..., f ( z n ))$ is a random vector with normal distribution. One way to generate them is using linear combinations of holomorphic functions whose coefficients are Gaussian random variables in $\mathbb{C}$ (or in $\mathbb{R}$ in special cases). For finding the zero set of a GAF we work on four isometric - invariant Hilbert spaces of analytic functions: the Fock space in $\mathbb{C}$, the finite space of polynomials in $\mathbb{S}^2$, the weighted Bergman space in $\mathbb{D}$ and the Paley - Wiener space. The first intensity determines the average of the distribution of the zero set of a GAF, and the Edelman - Kostlan formula gives an explicit expression of it. A result of uniqueness, called Calabi’s Rigidity, concludes that the first intensity determines the distribution of the zero set of a GAF. At the end, some examples made in C++ and gnuplot clarify the theory in these Hilbert spaces.