Overlap problems on the circle

Abstract

Consider a circle with perimeter N > 1 on which k < N segments of length 1 are sampled in an independent and identically distributed manner. In this paper we study the probability π (k,N) that these k segments do not overlap; the density ͣ(·) of the position of the disks on the circle is arbitrary (that is, it is not necessarily assumed uniform). Two scaling regimes are considered. In the first we set k≡ a√N, and it turns out that the probability of interest converges (N→ ∞) to an explicitly given positive constant that reflects the impact of the density ͣ(·). In the other regime k scales as aN, and the non-overlap probability decays essentially exponentially; we give the associated decay rate as the solution to a variational problem. Several additional ramifications are presented.