  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.
