Approach to Stationarity of the Bernoulli.Laplace Diffusion Model

Two urns initially contain r red balls and n . r black balls respectively. At each time epoch a ball is chosen randomly from each urn and the balls are switched. Effectively the same process arises in many other contexts, notably for a symmetric exclusion process and random walk on the Johnson graph. If Y(·) counts the number of black balls in the first urn then we give a direct asymptotic analysis of its transition probabilities to show that (when run at rate (n . r)/n in continuous time) for as n .., where . n denotes the equilibrium distribution of Y(·) and . . = 1 . . /. (1 . .). Thus for large n the transient probabilities approach their equilibrium values at time log n + log|. . | (.log n) in a particularly sharp manner. The same is true of the separation distance between the transient distribution and the equilibrium distribution. This is an explicit analysis of the so-called cut-off phenomenon associated with a wide variety of Markov chains.