pproach 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 π n denotes the equilibrium distribution of Y (·) and γ α = 1 – α / β (1 – β ). Thus for large n the transient probabilities approach their equilibrium values at time log n + log| γ α | (≦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.