STATISTICS OF BOSE SAMPLES FROM DIRICHLET PROPORTIONS
Thierry Huillet · 2006
To fix the background and notations, we shall first briefly revisit some aspects of the following Ewens-like randomized occupancy problem: as-sume distinguishable particles are to be placed at random into the cells of the unit interval which was previously broken into random pieces ac-cording to the (Poisson-)Dirichlet partitioning model. Particles being dis-tinguishable, the statistical structure of the problem can be understood within the Maxwell-Boltzmann setup. In this note, we shall address the following sampling problem of a dif-ferent nature: assume now that indistinguishable particles are to be placed at random within the cells with (Poisson-)Dirichlet distributed sizes. Then the statistical formalism to be used is the one of Bose-Einstein. We show that in the grand canonical ensemble, the Bose sampling procedure from (Poisson-)Dirichlet proportions is, to a large extent, amenable to exact analytic calculations. This concerns for example the full Bose occupancy distributions, the distribution of the number of distinct occupied frag-ments, the number of cells with a prescribed amount of particles. Using a grand canonical approach, a phase transition phenomenon is shown to take place provided the disorder parameter of the (Poisson-)Dirichlet par-tition is large enough; we describe this phase transition in some details.