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

Abstract

Consider N equally spaced points on a circle of circumferenceN. Pick atrandom n points out of N on this circle and consider the discrete random spacings between consecutive sampledpoints, turning clockwise. This defines in the first place a random partitioning ofN inton positive summands. Append then clockwise an arc of integral lengthk 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 parkingconfigurations, number and length of the gaps are addressed. For each value ofk, asymptotic resultsare presented when n, N both go to according to two different regimes. In the first thermodynamical regime , the occurrence of, say, covering and parking configurationsis exponentially rare in the whole admissible range of densityρ. We compute the rates from the equations of state. In the second one, theyare macroscopically frequent. These questions require some understandingof both the smallest and largest extreme summands in the partition ofN.We consider next an urn model whereN indistinguishable balls are assigned at random intoN distinguishable boxes. This urn model consists of a random partitioning model of integerN intoN non-negative summands.Given there are n non-empty boxes this gives back the original partitioning model ofN inton positive parts. Following this circle of ideas, a grand canonical balls in boxes approachis supplied, giving some insight into the multiplicities of the box occupancies.The random set model defines ak-nearest neighborrandom graph with N vertices and kn edges. We shall also briefly consider the covering problem in the context of a random graph model withN verticesand n (out-degree 1) edges whose endpoints are no longer bound to be neighbors. Inthe latter setup, connectivity is increased in that there exists a critical densityρc above which covering occurs with probability 1.