Random walks and the "Euclidean" association scheme in finite vector spaces
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2024
- DOI:
- 10.48550/arXiv.2401.04814
- arXiv:
- arXiv:2401.04814
- Bibcode:
- 2024arXiv240104814B
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Number Theory;
- 05E30;
- 05C90 (primary);
- 11L05;
- 11T23 (secondary)