Large Deviations for Intersections of Random Walks

Amine Asselah (1) , Bruno Schapira (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 I2M - Institut de Mathématiques de Marseille

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.

Mots clés: Random Walk; Range; Intersections; Large Deviations Principle

Domaines: Probabilités [math.PR]

Amine Asselah, Bruno Schapira. Large Deviations for Intersections of Random Walks. 2020. ⟨hal-02564305⟩