Discrete random Clark measures and associated inner functions

We study a class of random inner functions $\varphi$ 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 $\varphi$ is almost surely a Blaschke product.

We then investigate when $\varphi$ 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 $\varphi$.

Finally, we obtain quantitative estimates for the zeros of $\varphi$, proving that, in suitable regimes, their distribution satisfies summability conditions stronger than the classical Blaschke condition.