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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.

MSC:

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