## 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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