Discrete random Clark measures and associated inner functions

Abstract

We study a class of random inner functions $φ$ 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 $φ$ is almost surely a Blaschke product. We then investigate when $φ$ 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 $φ$. Finally, we obtain quantitative estimates for the zeros of $φ$, proving that, in suitable regimes, their distribution satisfies summability conditions stronger than the classical Blaschke condition.