Permanental Partition Models and Markovian Gibbs Structures

Abstract

We study both time-invariant and time-varying Gibbs distributions for configurations of particles into disjoint clusters. Specifically, we introduce and give some fundamental properties for a class of partition models, called permanental partition models, whose distributions are expressed in terms of the α-permanent of a similarity matrix parameter. We show that, in the time-invariant case, the permanental partition model is a refinement of the celebrated Pitman–Ewens distribution; whereas, in the time-varying case, the permanental model refines the Ewens cut-and-paste Markov chains (J. Appl. Probab. 43(3):778–791, 2011). By a special property of the α-permanent, the partition function can be computed exactly, allowing us to make several precise statements about this general model, including a characterization of exchangeable and consistent permanental models.