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arXiv · 2610.26640

Deviation probabilities for global events in two dimensions (I): independent excursions

Orphée Collin·Serguei Popov
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Abstract

We consider the field of local times generated by independent excursions of the two-dimensional simple random walk between the boundaries of two concentric disks, and study global events: informally, these are the events that have to occur everywhere (i.e., if such an event occurs on a set, then it also occurs on all its subsets). Examples include ``the set is completely covered'', ``the maximal local time on the set does not exceed a given level'', or ``the set does not contain a covered disk of a given radius''. For monotone global events, we show that the deviation probabilities (i.e., the probabilities that the event occurs with atypically few excursions, or does not occur with atypically many of them, or vice versa) can be obtained from the typical behaviour on different scales only, via a general decoupling argument based on soft local times.

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Orphée Collin, Serguei Popov. 2026-10-19. Deviation probabilities for global events in two dimensions (I): independent excursions. https://arxiv.org/abs/2610.26640

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