Capacity of the range of random walk: Moderate deviations in dimensions 4 and 5
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912694827745280 |
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| author | Adhikari, Arka Park, Jiyun |
| author_facet | Adhikari, Arka Park, Jiyun |
| contents | We prove a moderate deviation principle for the capacity of the range of random walk in $\mathbb{Z}^5$. Depending on the scale of deviation, we get two different regimes. We observe Gaussian tails when the deviation scale is smaller than $n^{1/2} (\log n)^{3/4}$. Otherwise, we get non-Gaussian tails with a constant arising from a generalized Gagliardo-Nirenberg inequality. This is analogous to the behavior of the volume of the random walk range in $\mathbb{Z}^3$. Our methods can also be applied to the $d = 4$ case to prove the moderate deviation principle in almost the full range of interest. This extends the work of Okada and the first author \cite{AdhikariOkada2023}, where they showed moderate deviations up to a deviation scale of $\log \log n$ times the standard deviation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_05585 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Capacity of the range of random walk: Moderate deviations in dimensions 4 and 5 Adhikari, Arka Park, Jiyun Probability 60F10, 60G50 We prove a moderate deviation principle for the capacity of the range of random walk in $\mathbb{Z}^5$. Depending on the scale of deviation, we get two different regimes. We observe Gaussian tails when the deviation scale is smaller than $n^{1/2} (\log n)^{3/4}$. Otherwise, we get non-Gaussian tails with a constant arising from a generalized Gagliardo-Nirenberg inequality. This is analogous to the behavior of the volume of the random walk range in $\mathbb{Z}^3$. Our methods can also be applied to the $d = 4$ case to prove the moderate deviation principle in almost the full range of interest. This extends the work of Okada and the first author \cite{AdhikariOkada2023}, where they showed moderate deviations up to a deviation scale of $\log \log n$ times the standard deviation. |
| title | Capacity of the range of random walk: Moderate deviations in dimensions 4 and 5 |
| topic | Probability 60F10, 60G50 |
| url | https://arxiv.org/abs/2507.05585 |