On Pólya's random walk constants
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917605403525120 |
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| author | Gaunt, Robert E. Nadarajah, Saralees Pogány, Tibor K. |
| author_facet | Gaunt, Robert E. Nadarajah, Saralees Pogány, Tibor K. |
| contents | A celebrated result in probability theory is that a simple symmetric random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ is recurrent for $d=1,2$ and transient for $d\geq 3$. In this note, we derive a closed-form expression, in terms of the Lauricella function $F_C$, for the return probability for all $d\geq3$. Previously, a closed-form formula had only been available for $d=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11326 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Pólya's random walk constants Gaunt, Robert E. Nadarajah, Saralees Pogány, Tibor K. Probability Classical Analysis and ODEs 60G50, 33C65 A celebrated result in probability theory is that a simple symmetric random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ is recurrent for $d=1,2$ and transient for $d\geq 3$. In this note, we derive a closed-form expression, in terms of the Lauricella function $F_C$, for the return probability for all $d\geq3$. Previously, a closed-form formula had only been available for $d=3$. |
| title | On Pólya's random walk constants |
| topic | Probability Classical Analysis and ODEs 60G50, 33C65 |
| url | https://arxiv.org/abs/2311.11326 |