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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.04814 |
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| _version_ | 1866917565185392640 |
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| author | Brittenham, Charles Pakianathan, Jonathan |
| author_facet | Brittenham, Charles Pakianathan, Jonathan |
| contents | In this paper, we provide an application to the random distance-$t$ walk in finite planes and derive asymptotic formulas (as $q \to \infty$) for the probability of return to start point after $\ell$ steps based on the "vertical" equidistribution of Kloosterman sums established by N. Katz. The application of these deep results from number theory allow a determination of the second order terms in the answers that simpler spectral gap/mixing rate methods do not. This work relies on a "Euclidean" association scheme studied in prior work of W.M.Kwok, E. Bannai, O. Shimabukuro and H. Tanaka. We also provide a self-contained discussion of the P-matrix and intersection numbers of this scheme for convenience in our application as well as a more explicit form for the intersection numbers in the planar case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04814 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random walks and the "Euclidean" association scheme in finite vector spaces Brittenham, Charles Pakianathan, Jonathan Combinatorics Number Theory 05E30, 05C90 (primary), 11L05, 11T23 (secondary) In this paper, we provide an application to the random distance-$t$ walk in finite planes and derive asymptotic formulas (as $q \to \infty$) for the probability of return to start point after $\ell$ steps based on the "vertical" equidistribution of Kloosterman sums established by N. Katz. The application of these deep results from number theory allow a determination of the second order terms in the answers that simpler spectral gap/mixing rate methods do not. This work relies on a "Euclidean" association scheme studied in prior work of W.M.Kwok, E. Bannai, O. Shimabukuro and H. Tanaka. We also provide a self-contained discussion of the P-matrix and intersection numbers of this scheme for convenience in our application as well as a more explicit form for the intersection numbers in the planar case. |
| title | Random walks and the "Euclidean" association scheme in finite vector spaces |
| topic | Combinatorics Number Theory 05E30, 05C90 (primary), 11L05, 11T23 (secondary) |
| url | https://arxiv.org/abs/2401.04814 |