Mikhail Sodin: The Bargmann--Fock excess of a randomly perturbed lattice

Abstract:  A classical theorem of Perelomov states that the square lattice \(\mathbb{Z}^2\), with one point removed, is a uniqueness set for the Bargmann–Fock space, while removing two points from \(\mathbb{Z}^2\) produces the zero set of a non-zero Bargmann–Fock function.

We consider the same question for the random point process obtained by perturbing the points of \(\mathbb{Z}^2\) by independent complex Gaussian variables with standard deviation \(b\). We show that the Bargmann–Fock excess of this process almost surely equals 1 for small values of \(b\). For larger \(b\), it is an explicitly computed piecewise constant function of \(b\), growing asymptotically linearly with \(b\).

The talk is based on joint work with Alexander Borichev.