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

Charles Brittenham, Jonathan Pakianathan

Published 2024-01-09Version 1

In this paper, we study an association scheme derived from the "Euclidean" quadratic form on finite vector spaces over finite fields of odd prime order $q$. We calculate the change of basis matrices $P$, $Q$ between the geometric and spectral basis of the corresponding Bose-Messner algebra as well as the intersection numbers of the scheme. As an application we study 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.