Branching capacity of a random walk in $\mathbb Z^5$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| author | Bai, Tianyi Delmas, Jean-François Hu, Yueyun |
| author_facet | Bai, Tianyi Delmas, Jean-François Hu, Yueyun |
| contents | We are interested in the branching capacity of the range of a random walk in $\mathbb Z^d$.Schapira [28] has recently obtained precise asymptotics in the case $d\ge 6$ and has demonstrated a transition at dimension $d=6$. We study the case $d=5$ and prove that the renormalized branching capacity converges in law to the Brownian snake capacity of the range of a Brownian motion. The main step in the proof relies on studying the intersection probability between the range of a critical Branching random walk and that of a random walk, which is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19447 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Branching capacity of a random walk in $\mathbb Z^5$ Bai, Tianyi Delmas, Jean-François Hu, Yueyun Probability 60F05, 60J45, 60J80 We are interested in the branching capacity of the range of a random walk in $\mathbb Z^d$.Schapira [28] has recently obtained precise asymptotics in the case $d\ge 6$ and has demonstrated a transition at dimension $d=6$. We study the case $d=5$ and prove that the renormalized branching capacity converges in law to the Brownian snake capacity of the range of a Brownian motion. The main step in the proof relies on studying the intersection probability between the range of a critical Branching random walk and that of a random walk, which is of independent interest. |
| title | Branching capacity of a random walk in $\mathbb Z^5$ |
| topic | Probability 60F05, 60J45, 60J80 |
| url | https://arxiv.org/abs/2404.19447 |