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- W2152509544 abstract "We study the range $R_n$ of a random walk on the $d$-dimensional lattice $mathbb{Z}^d$ indexed by a random tree with $n$ vertices. Under the assumption that the random walk is centered and has finite fourth moments, we prove in dimension $dleq3$ that $n^{-d/4}R_n$ converges in distribution to the Lebesgue measure of the support of the integrated super-Brownian excursion (ISE). An auxiliary result shows that the suitably rescaled local times of the tree-indexed random walk converge in distribution to the density process of ISE. We obtain similar results for the range of critical branching random walk in $mathbb{Z}^d$, $dleq3$. As an intermediate estimate, we get exact asymptotics for the probability that a critical branching random walk starting with a single particle at the origin hits a distant point. The results of the present article complement those derived in higher dimensions in our earlier work." @default.
- W2152509544 created "2016-06-24" @default.
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- W2152509544 date "2015-09-01" @default.
- W2152509544 modified "2023-10-17" @default.
- W2152509544 title "The range of tree-indexed random walk in low dimensions" @default.
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- W2152509544 doi "https://doi.org/10.1214/14-aop947" @default.
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