id: 00672304
dt: j
an: 00672304
au: Donnelly, Peter; Lloyd, Peter; Sudbury, Aidan
ti: Approach to stationarity of the Bernoulli-Laplace diffusion model.
so: Adv. Appl. Probab. 26, No.3, 715-727 (1994).
py: 1994
pu: Applied Probability Trust, Sheffield
la: EN
cc: 
ut: distance transitive graph; Eberlein polynomials; exclusion process; Johnson
    graph; transition probabilities; separation distance; transient
    distribution; equilibrium distribution; cut-off phenomenon
ci: 
li: doi:10.2307/1427817
ab: Summary: 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 $Y(\cdot)$ 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 $j= αn+ o(n)$, $r= βn+ o(n)$,
    $0\leqα\leqβ\leq {1\over 2}$, $β>0$, $${\bold P} (Y(\log n+c)= j)/
    π\sb n(j)\to \exp (γ\sb αe\sp{-c}) \quad \text{as} \quad
    n\to\infty,$$ where $π\sb n$ denotes the equilibrium distribution of
    $Y(\cdot)$ and $γ\sb α= 1-α/ β(1- β)$. Thus for large $n$ the
    transient probabilities approach their equilibrium values at time $\log
    n+ \log \vert γ\sb α\vert$ $(\leq\log n)$ 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.
rv: