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Sequential random displacements of points in an interval

An array of points {Z 1, Z 2, …, Zn– 1} in the interval V = [0, L], is such that 0 ≦ Z 1, ≦ Z 2 ≦ … ≦ Z n–1, ≦ L. One of the points is chosen at random (Zk , say, with probability p k) and displaced to a new position within the interval [Zk –1, Zk + 1], the position again chosen at random according to a probability distribution Gk. We derive some results concerning the limiting distribution of the array after a succession of such displacements. If Gk is a uniform distribution, it appears that the number of displacements necessary to open up a gap between at least one pair of adjacent points of size at least γ is O(ρ n ), n →∞, where ρ = L/(L – γ).