Geodesic
Random walks and the “Euclidean” association scheme in finite vector spaces
Canadian journal of mathematics, Tome 77 (2025) no. 5, pp. 1664-1685

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DOI

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. 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 computation 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.
DOI : 10.4153/S0008414X24000518
Mots-clés : Association schemes, Kloosterman sums, random walks 
Brittenham, Charles; Pakianathan, Jonathan. Random walks and the “Euclidean” association scheme in finite vector spaces. Canadian journal of mathematics, Tome 77 (2025) no. 5, pp. 1664-1685. doi: 10.4153/S0008414X24000518
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     title = {Random walks and the {{\textquotedblleft}Euclidean{\textquotedblright}} association scheme in finite vector spaces},
     journal = {Canadian journal of mathematics},
     pages = {1664--1685},
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