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arXiv · 2607.08592

Discrete random Clark measures and associated inner functions

Carlo Bellavita·Nikolaos Chalmoukis·Giuseppe Lamberti
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Abstract

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.

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Carlo Bellavita, Nikolaos Chalmoukis, Giuseppe Lamberti. 2026-07-09. Discrete random Clark measures and associated inner functions. https://arxiv.org/abs/2607.08592

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