Large Deviations for Intersections of Random Walks

2020/05/06 by Amine Asselah, Bruno Schapira, Asselah, Amine +1 · 1 citation
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · doi:10.48550/arxiv.2005.02735

openalex publication_date 2020/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Sinaï [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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