Rates of convergence of random walk on distance regular graphs

When run on any non-bipartite q-distance regular graph from a family containing graphs of arbitrarily large diameter d, we show that d steps are necessary and sufficient to drive simple random walk to the uniform distribution in total variation distance, and that a sharp cutoff phenomenon occurs. For most examples, we determine the set on which the variation distance is achieved, and the precise rate at which it decays.The upper bound argument uses spectral methods – combining the usual Cauchy-Schwarz bound on variation distance with a bound on the tail probability of a first-hitting time, derived from its generating function.