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arXiv · 1307.5221

The range of tree-indexed random walk

Jean-François Le Gall·Shen Lin
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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 R(n)/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 branching random walk when the initial size of the population tends to infinity.

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Jean-François Le Gall, Shen Lin. 2013-07-19. The range of tree-indexed random walk. https://arxiv.org/abs/1307.5221

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