Random Walks and the ”euclidean” Association Scheme in Finite Vector Spaces

In this paper, we provide an application to the random distance- t walk in finite planes and derive asymptotic formulas (as q → ∞ ) for the probability of return to start point after 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.