Covering by random intervals and one-dimensional continuum percolation

Abstract

A brief historical introduction is given to the problem of covering a line by random overlapping intervals. The problem for equal intervals was first solved by Whitworth in the 1890s. A brief resume is given of his solution. The advantages of the present author's approach, which uses a Poisson process, are outlined, and a solution is derived by Laplace transforms. The method of Hammersley for dealing with a stochastic distribution of intervals is described, and a solution can still be derived by Laplace transforms. The asymptotic behavior as the line becomes long is calculated and is related to the one-dimensional continuum percolation problem. It is shown that as long as the mean interval size is finite, the probability of complete coverage decays exponentially, so that the critical percolation probabilityp _c =1. However, as soon as the mean interval size becomes infinite, the critical percolation probabilityp _c switches to 0. This is in accord with previous results for a lattice model by Chinese workers, but differs from those of Schulman. A possible reason for the discrepancy is a difference in boundary conditions.