Large Deviations for Intersections of Random Walks
Probability
2020-05-07 v1
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.
Cite
@article{arxiv.2005.02735,
title = {Large Deviations for Intersections of Random Walks},
author = {Amine Asselah and Bruno Schapira},
journal= {arXiv preprint arXiv:2005.02735},
year = {2020}
}