Abstract:
Consider a branching random walk $(V_u)_{u\in \mathcal T^{\mathrm{IGW}}}$ in $\mathbb Z^d$ with the genealogy tree $\mathcal T^{\mathrm{IGW}}$ formed by a sequence of i.i.d. critical Galton–Watson trees. Let $R_n$ be the set of points in $\mathbb Z^d$ visited by $(V_u)$ when the index $u$ explores the first $n$ subtrees in $\mathcal T^{\mathrm{IGW}}$. Our main result states that for $d\in \{3,4,5\}$, the capacity of $R_n$ is almost surely equal to $n^{(d-2)/{2}+o(1)}$ as $n\to \infty $.
Keywords:branching random walk, tree-indexed random walk, capacity.
Citation:
Tianyi Bai, Yueyun Hu, “Capacity of the Range of Branching Random Walks in Low Dimensions”, Branching Processes and Related Topics, Collected papers. On the occasion of the 75th birthday of Andrei Mikhailovich Zubkov and 70th birthday of Vladimir Alekseevich Vatutin, Trudy Mat. Inst. Steklova, 316, Steklov Math. Inst., Moscow, 2022, 32–46; Proc. Steklov Inst. Math., 316 (2022), 26–39
\Bibitem{BaiHu22}
\by Tianyi~Bai, Yueyun~Hu
\paper Capacity of the Range of Branching Random Walks in Low Dimensions
\inbook Branching Processes and Related Topics
\bookinfo Collected papers. On the occasion of the 75th birthday of Andrei Mikhailovich Zubkov and 70th birthday of Vladimir Alekseevich Vatutin
\serial Trudy Mat. Inst. Steklova
\yr 2022
\vol 316
\pages 32--46
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4217}
\crossref{https://doi.org/10.4213/tm4217}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2022
\vol 316
\pages 26--39
\crossref{https://doi.org/10.1134/S0081543822010047}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85140789904}
Linking options:
https://www.mathnet.ru/eng/tm4217
https://doi.org/10.4213/tm4217
https://www.mathnet.ru/eng/tm/v316/p32
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