Random walks veering left. (English) Zbl 1296.60117

The authors study coupled random walks in the plane such that, at each step, the walks change direction by a uniform random angle plus an extra deterministic angle \(\theta\). Let such a walk be called \(S_n(\theta)\). By the central limit theorem, \(|S_n(\theta)|\) is, almost surely, of the order \(\sqrt{n}\). The authors show that the set of “exceptional angles”, i.e., angles for which \(|S_n (\theta)| \) exceeds a certain larger threshold of the form \(\sqrt{2\alpha n \log n} \) infinitely often, is non-empty and even may have a positive Hausdorff dimension \((1 - \alpha) \vee 0 \). The motivation for the model comes from random matrix theory. The techniques developed are also used to study the boundary behavior of Gaussian analytic functions.


60G50 Sums of independent random variables; random walks
60B20 Random matrices (probabilistic aspects)
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