THE RANGE OF TREE-INDEXED RANDOM WALK. (10th September 2014)

• Record Type: Journal Article
• Authors: Le Gall, Jean-François; Lin, Shen
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 15:Number 2(2016:Apr.)
• Page Start: 271
• Page End: 317
• Publication Date: 2014-09-10
• Subjects: 60G50, -- 60J80; tree-indexed random walk, -- range, -- discrete snake, -- branching random walk, -- subadditive ergodic theorem; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S1474748014000280
• Languages: English
• ISSNs: 1474-7480

Abstract

Abstract : We provide asymptotics for the range $R_{n}$ of a random walk on the $d$ -dimensional lattice indexed by a random tree with $n$ vertices. Using Kingman's subadditive ergodic theorem, we prove under general assumptions that $n^{-1}R_{n}$ converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension $4$, and in the case of a symmetric random walk with exponential moments, we prove that $R_{n}$ grows like $n/\!\log n$ . We apply our results to asymptotics for the range of a branching random walk when the initial size of the population tends to infinity.