1 results
THE RANGE OF TREE-INDEXED RANDOM WALK
- Part of
-
- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 15 / Issue 2 / April 2016
- Published online by Cambridge University Press:
- 10 September 2014, pp. 271-317
- Print publication:
- April 2016
-
- Article
- Export citation
-
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.
