Geodesic
Capacity of the Range of Branching Random Walks in Low Dimensions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 32-46.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{TM_2022_316_a3,
     author = {Tianyi Bai and Yueyun Hu},
     title = {Capacity of the {Range} of {Branching} {Random} {Walks} in {Low} {Dimensions}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {32--46},
     publisher = {mathdoc},
     volume = {316},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2022_316_a3/}
}
TY  - JOUR
AU  - Tianyi Bai
AU  - Yueyun Hu
TI  - Capacity of the Range of Branching Random Walks in Low Dimensions
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2022
SP  - 32
EP  - 46
VL  - 316
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2022_316_a3/
LA  - ru
ID  - TM_2022_316_a3
ER  - 
%0 Journal Article
%A Tianyi Bai
%A Yueyun Hu
%T Capacity of the Range of Branching Random Walks in Low Dimensions
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2022
%P 32-46
%V 316
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2022_316_a3/
%G ru
%F TM_2022_316_a3
Tianyi Bai; Yueyun Hu. Capacity of the Range of Branching Random Walks in Low Dimensions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 32-46. http://geodesic.mathdoc.fr/item/TM_2022_316_a3/

[1] Asselah A., Schapira B., Sousi P., “Capacity of the range of random walk on $\mathbb Z^d$”, Trans. Amer. Math. Soc., 370:11 (2018), 7627–7645 | DOI | MR | Zbl

[2] Bai T., Wan Y., “Capacity of the range of tree-indexed random walk”, Ann. Appl. Probab. (to appear)

[3] Csáki E., “On the lower limits of maxima and minima of Wiener process and partial sums”, Z. Wahrscheinlichkeitstheor. verw. Geb., 43 (1978), 205–221 | DOI | MR | Zbl

[4] Duquesne Th., Le Gall J.-F., Random trees, Lévy processes and spatial branching processes, Astérisque, 281, Soc. math. France, Paris, 2002 | MR

[5] Kersting G., Vatutin V., Discrete time branching processes in random environment, J. Wiley Sons, Hoboken, NJ, 2017 | Zbl

[6] Lawler G.F., Limic V., Random walk: A modern introduction, Cambridge Stud. Adv. Math., 123, Cambridge Univ. Press, Cambridge, 2010 | MR | Zbl

[7] Le Gall J.-F., Lin S., “The range of tree-indexed random walk in low dimensions”, Ann. Probab., 43:5 (2015), 2701–2728 | MR | Zbl

[8] Le Gall J.-F., Lin S., “The range of tree-indexed random walk”, J. Inst. Math. Jussieu, 15:2 (2016), 271–317 | DOI | MR | Zbl

[9] Petrov V.V., Limit theorems of probability theory: Sequences of independent random variables, Oxford Stud. Probab., 4, Clarendon Press, Oxford, 1995 | MR | Zbl

[10] Zhu Q., “On the critical branching random walk. III: The critical dimension”, Ann. Inst. Henri Poincaré. Probab. stat., 57:1 (2021), 73–93 | MR | Zbl