Sequential random displacements of points in an interval

Abstract

An array of points {Z1, Z2, …, Zn–1} in the interval V = [0, L], is such that 0 ≦ Z1, ≦ Z2 ≦ … ≦ Zn–1, ≦ L. One of the points is chosen at random (Zk, say, with probability pk) 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).