Moderate deviations for the range of a transient random walk: path concentration

Asselah, Amine; Schapira, Bruno

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 3, pp. 755-786.

Nous étudions les déviations qui réduisent la frontière du support d'une marche transiente sur le réseau euclidien. Nous décrivons en particulier une stratégie optimale pour réduire la frontière du support. Les techniques employées s'appliquent aussi bien au volume du support lui-même, et fournissent des énoncés mathématiques qui illustrent l'image du « fromage suisse » de Bolthausen, van den Berg et den Hollander.

We study downward deviations of the boundary of the range of a transient walk on the Euclidean lattice. We describe the optimal strategy adopted by the walk in order to shrink the boundary of its range. The technics we develop apply equally well to the range, and provide pathwise statements for the Swiss cheese picture of Bolthausen, van den Berg and den Hollander [7].

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