Large Deviations for Intersections of Random Walks

Amine Asselah; Bruno Schapira

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arXiv · 2005.02735

Large Deviations for Intersections of Random Walks

Amine Asselah·Bruno Schapira

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Abstract

We prove a Large Deviations Principle for the number of intersections of two independent infinite-time ranges in dimension five and more, improving upon the moment bounds of Khanin, Mazel, Shlosman and Sina{\"i} [KMSS94]. This settles, in the discrete setting, a conjecture of van den Berg, Bolthausen and den Hollander [BBH04], who analyzed this question for the Wiener sausage in finite-time horizon. The proof builds on their result (which was resumed in the discrete setting by Phetpradap [Phet12]), and combines it with a series of tools that were developed in recent works of the authors [AS17, AS19a, AS20]. Moreover, we show that most of the intersection occurs in a single box where both walks realize an occupation density of order one.

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Amine Asselah, Bruno Schapira. 2020-05-06. Large Deviations for Intersections of Random Walks. https://arxiv.org/abs/2005.02735

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