Abstract
Consider a branching random walk \((V_u)_{u\in\mathcal T^{\text{IGW}}}\) in \(\mathbb Z^d\) with the genealogy tree \(\mathcal T^{\text{IGW}}\) 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 \(\mathcal T^{\text{IGW}}\). 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 \(n\to\infty\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Asselah, B. Schapira, and P. Sousi, “Capacity of the range of random walk on \(\mathbb Z^d\),” Trans. Am. Math. Soc. 370 (11), 7627–7645 (2018).
T. Bai and Y. Wan, “Capacity of the range of tree-indexed random walk,” Ann. Appl. Probab. (in press); arXiv: 2004.06018 [math.PR].
E. Csáki, “On the lower limits of maxima and minima of Wiener process and partial sums,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 43, 205–221 (1978).
Th. Duquesne and J.-F. Le Gall, Random Trees, Lévy Processes and Spatial Branching Processes (Soc. Math. France, Paris, 2002), Astérisque 281.
G. Kersting and V. Vatutin, Discrete Time Branching Processes in Random Environment (J. Wiley & Sons, Hoboken, NJ, 2017).
G. F. Lawler and V. Limic, Random Walk: A Modern Introduction (Cambridge Univ. Press, Cambridge, 2010), Cambridge Stud. Adv. Math. 123.
J.-F. Le Gall and S. Lin, “The range of tree-indexed random walk in low dimensions,” Ann. Probab. 43 (5), 2701–2728 (2015).
J.-F. Le Gall and S. Lin, “The range of tree-indexed random walk,” J. Inst. Math. Jussieu 15 (2), 271–317 (2016).
V. V. Petrov, Limit Theorems of Probability Theory: Sequences of Independent Random Variables (Clarendon Press, Oxford, 1995), Oxford Stud. Probab. 4.
Q. Zhu, “On the critical branching random walk. III: The critical dimension,” Ann. Inst. Henri Poincaré, Probab. Stat. 57 (1), 73–93 (2021).
Acknowledgments
We are grateful to an anonymous referee for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Vol. 316, pp. 32–46 https://doi.org/10.4213/tm4217.
Dedicated to the 75th anniversary of Professor Andrei M. Zubkov and the 70th anniversary of Professor Vladimir A. Vatutin
Rights and permissions
About this article
Cite this article
Bai, T., Hu, Y. Capacity of the Range of Branching Random Walks in Low Dimensions. Proc. Steklov Inst. Math. 316, 26–39 (2022). https://doi.org/10.1134/S0081543822010047
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543822010047