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In this paper, we provide an application to the random distance-$t$ walk infinite planes and derive asymptotic formulas (as $q \to \infty$) for theprobability of return to start point after $\ell$ steps based on the "vertical"equidistribution of Kloosterman sums established by N. Katz. The application ofthese deep results from number theory allow a determination of the second orderterms in the answers that simpler spectral gap/mixing rate methods do not. Thiswork relies on a "Euclidean" association scheme studied in prior work ofW.M.Kwok, E. Bannai, O. Shimabukuro and H. Tanaka. We also provide aself-contained discussion of the P-matrix and intersection numbers of thisscheme for convenience in our application as well as a more explicit form forthe intersection numbers in the planar case.

Random walks and the "Euclidean" association scheme in finite vector spaces