Geodesic
Capacity of the Range of Branching Random Walks in Low Dimensions
Informatics and Automation, Branching Processes and Related Topics, Tome 316 (2022), pp. 32-46

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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. 
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     author = {Tianyi Bai and Yueyun Hu},
     title = {Capacity of the {Range} of {Branching} {Random} {Walks} in {Low} {Dimensions}},
     journal = {Informatics and Automation},
     pages = {32--46},
     publisher = {mathdoc},
     volume = {316},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2022_316_a3/}
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Tianyi Bai; Yueyun Hu. Capacity of the Range of Branching Random Walks in Low Dimensions. Informatics and Automation, Branching Processes and Related Topics, Tome 316 (2022), pp. 32-46. http://geodesic.mathdoc.fr/item/TRSPY_2022_316_a3/