Approach to Stationarity of the Bernoulli–Laplace Diffusion Model

<jats:p>Two urns initially contain <jats:italic>r</jats:italic> red balls and <jats:italic>n – r</jats:italic> 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 <jats:italic>Y</jats:italic>(·) 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 (<jats:italic>n – r</jats:italic>)/<jats:italic>n</jats:italic> in continuous time) for <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0001867800026513_inline1" /> as <jats:italic>n</jats:italic> →∞, where <jats:italic>π <jats:sub>n</jats:sub></jats:italic> denotes the equilibrium distribution of <jats:italic>Y</jats:italic>(·) and <jats:italic>γ <jats:sub>α</jats:sub></jats:italic> = 1 – <jats:italic>α</jats:italic> /<jats:italic>β</jats:italic> (1 – <jats:italic>β</jats:italic>). Thus for large <jats:italic>n</jats:italic> the transient probabilities approach their equilibrium values at time log <jats:italic>n</jats:italic> + log|<jats:italic>γ <jats:sub>α</jats:sub></jats:italic> | (≦log <jats:italic>n</jats:italic>) 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.</jats:p>