2022/04/14 by Benedikt Stufler, Stufler, Benedikt · 1 citation
Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2204.06982
openalex publication_date 2022/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (2018, Random Structures & Algorithms) 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.