Approach to Stationarity of the Bernoulli–Laplace Diffusion Model

Two urns initially containrred balls andn – rblack 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. IfY(·) 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)/nin continuous time) forasn→∞, whereπndenotes the equilibrium distribution ofY(·) andγα= 1 –α/β(1 –β). Thus for largenthe transient probabilities approach their equilibrium values at time logn+ log|γα| (≦logn) 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.