A Bose–Einstein approach to the random partitioning of an integer

Consider N equally spaced points on a circle of circumference N. Pick at random n points out of N on this circle and consider the discrete random spacings between consecutive sampled points, turning clockwise. This defines in the first place a random partitioning of N into n positive summands. Append then clockwise an arc of integral length k to each such sampled point, ending up with a discrete random set on the circle. Questions such as the evaluation of the probability of random covering or parking configurations, number and length of the gaps are addressed. For each value of k, asymptotic results are presented when n, N both go to according to two different regimes. In the first thermodynamical regime , the occurrence of, say, covering and parking configurations is exponentially rare in the whole admissible range of density ρ. We compute the rates from the equations of state. In the second one, they are macroscopically frequent. These questions require some understanding of both the smallest and largest extreme summands in the partition of N. We consider next an urn model where N indistinguishable balls are assigned at random into N distinguishable boxes. This urn model consists of a random partitioning model of integer N into N non-negative summands. Given there are n non-empty boxes this gives back the original partitioning model of N into n positive parts. Following this circle of ideas, a grand canonical balls in boxes approach is supplied, giving some insight into the multiplicities of the box occupancies. The random set model defines a k-nearest neighbor random graph with N vertices and kn edges. We shall also briefly consider the covering problem in the context of a random graph model with N vertices and n (out-degree 1) edges whose endpoints are no longer bound to be neighbors. In the latter setup, connectivity is increased in that there exists a critical density ρc above which covering occurs with probability 1.