Consider a reproducing kernel Hilbert space $H$ of functions defined on a domain $\Omega$, with reproducing kernel $K$. Given a sequence $\Lambda$ on $\Omega$, it is possible to associate to it a measure $\mu_\Lambda$. Understanding when this measure is a Carleson measure for $H$ is equivalent to know when the family of normalized reproducing kernels evaluated in the points of the sequence $\Lambda$ forms a Bessel system for $H$.

Our work focuses on understanding when the measure $\mu_\Lambda$ is a Carleson measure when $H$ is the Hardy space in the polydisc. While such measures are well understood and characterized by the ’one-box condition’ for the Hardy space in one complex variable, in the polydisc the situation is more complex, and this condition does not directly generalize.

In this scenario is then useful to use a different approach. We consider random sequences with prescribed radii and we provide a 0-1 law that describes when the measure $\mu_\Lambda$ is a Carleson measure almost surely. Using tools from random matrix theory, specifically a matrix Chernoff's inequality, we actually give the 0-1 law ensuring that the Gramian associated to the sequence $\Lambda$ is almost surely bounded.

This is a joint work with N. Chalmoukis and A. Dayan.