Random Walk, Local times, and subsets maximizing capacity

Amine Asselah (1) , Bruno Schapira (2)

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

We prove that in any finite set, in Z d with d ≥ 3, there is a subset whose capacity and volume are both of the same order as the capacity of the initial set. This implies some optimal estimates on probabilities of various folding events of a random walk. For instance, knowing that a random walk folds into an atypical high density region, we show that this random region is most likely ball-like.

Mots clés: Random Walk; Local times; Capacity; Range MSC 2010 subject classifications Primary 60F05; 60G50

Domaines: Probabilités [math.PR]

Amine Asselah, Bruno Schapira. Random Walk, Local times, and subsets maximizing capacity. 2020. ⟨hal-02499113v1⟩