Summary: Let \(X\) be the collection of \(k\)-dimensional subspaces of an \(n\)- dimensional vector space \(V_ n\) over \(GF(q)\). A metric may be defined on \(X\) by letting \(d(W_ k, V_ k) = k - \dim (W_ k \cap V_ k)\) for \(W_ k\), \(V_ k \in X\). The nearest neighbor random walk on \(X\) starts at an initial fixed point \(Z_ k\) of \(X\) and, from wherever it finds itself, moves with the same probability to any of the points of \(X\) at distance one from it. We analyze the rate of convergence of the nearest neighbor random walk on \(X\) to its stationary distribution. The argument involves lifting the process on \(X\) to a random walk on \(GL_ n (GF(q))\) and using the Fourier transform and \(q\)-Hahn polynomials.