Gibbs partitions: A comprehensive phase diagram

Abstract

We study Gibbs partition models, also known as composition schemes. Our main results comprehensively describe their phase diagram, including a phase transition from the convergent case described in Stufler (Random Structures Algorithms 53 (2018) 537–558) to a new dense regime characterized by a linear number of components with fluctuations of smaller order quantified by an α-stable law for 1 < α ≤ 2. We prove a functional scaling limit for a process whose jumps correspond to the component sizes and discuss applications to extremal component sizes. At the transition we observe a mixture of the two asymptotic shapes. We also treat extended composition schemes and prove a local limit theorem in a dilute regime with the limiting law being related to an α-stable law for 0 < α < 1. We describe the asymptotic size of the largest components via a point process limit.