Discrete random Clark measures and associated inner functions

Abstract

We study a class of random inner functions $\ddot{I}†$ whose Clark measure at $1$ is the weighted sum of point masses supported on independent uniformly distributed points of $\mathbb{T}$. Our first result shows that $\ddot{I}†$ is almost surely a Blaschke product. We then investigate when $\ddot{I}†$ admits angular derivative almost surely and we provide a $0-1$ law. These conditions have a direct interpretation in terms of the other Clark measures associated with $\ddot{I}†$. Finally, we obtain quantitative estimates for the zeros of $\ddot{I}†$, proving that, in suitable regimes, their distribution satisfies summability conditions stronger than the classical Blaschke condition.

22 pages