Mikhail Sodin

The Bargmann-Fock excess for the randomly perturbed lattice

A classical theorem of Perelomov states that the standard square lattice,
with one point removed, is a uniqueness set for the Bargmann-Fock space
of entire functions, while removing two points produces the zero set of
a nonzero Bargmann-Fock function.

In ongoing work with Alexander Borichev, we consider the same question
for the random point process obtained by perturbing the square lattice
by independent complex Gaussian variables. Somewhat unexpectedly,
the answer depends on the size of the perturbation and — probably
more unexpectedly — relies heavily on the theory of log-correlated fields.

If time permits, I will also discuss the answer to a similar question concerning
the excess of the sine-beta point process in the Paley-Wiener space of entire
function, obtained in work in progress with Paul Bourgade and Ofer Zeitouni.