APPROACH TO STATIONARITY OF THE BERNOULLI-LAPLACE DIFFUSION-MODEL

Two urns initially contain r red balls and n - r black balls respectiv ely. At each time epoch a ball is chosen randomly from each urn and th e balls are switched. Effectively the same process arises in many othe r 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 t ime) for j = alphan + o(n), r = betan + o(n), 0 less-than-or-equal-to alpha less-than-or-equal-to beta less-than-or-equal-to 1/2, beta > 0, P(Y(log n + c) = j)/pi(n)(j) --> exp (gamma(alpha)e(-c)) as n --> infi nity, where pi(n) denotes the equilibrium distribution of Y(.) and gam ma(alpha) = 1 - alpha/beta(1 - beta). Thus for large n the transient p robabilities approach their equilibrium values at time log n + log\gam ma(alpha)\ (less-than-or-equal-to log n) in a particularly sharp manne r. The same is true of the separation distance between the transient d istribution and the equilibrium distribution. This is an explicit anal ysis of the so-called cut-off phenomenon associated with a wide variet y of Markov chains.