Title : Gaussian analytic functions and operator symbols of Dirichlet type
Abstract: If we consider the classical Dirichlet space on the unit disk, there is a canonical
way to associate a random Gaussian function to it. Next, we take two copies of such a
Gaussian analytic function, but allow for dependence between them. This gives rise to two correlations, the holomorphic and conjugate-holomorphic correlations. We study the holomorphic correlation, which is induced by a contraction on L^2 on the disk.
An interesting choice of contraction is the classical Grunsky operator, and then the holomorphic correlation (=the Dirichlet operator symbol) connects with basic issues in the theory of conformal mappings. Such operator symbols for Grunsky may be characterised as solutions to a certain nonlinear wave equation. We also look at the asymptotic variance (introduced by McMullen in a dynamical context) associated with the Dirichlet symbol and find that the answer for general contractions and for Grunsky operators is different.