Large deviations analysis for random combinatorial partitions with counter terms

Abstract

Abstract In this paper, we study various models for random combinatorial partitions using large deviation analysis for diverging scale of the reference process. The large deviation rate functions are normalised limiting free energies and the main focus is to study their minimiser for various Gibbsian ensembles with respect to the reference measure which is a probabilistic version of the ideal Bose gas. Scaling limits of similar models have been studied recently (Fatkullin and Slastikov 2018 arXiv:1801.00812v2; Fatkullin and Xue 2021 J. Stat. Phys. 183 22) going back to (Vershik 1996 Func. Anal. Appl. 30 90–105). After studying the reference model, we provide a complete analysis of two mean field models, one of which is well-know (Benfatto et al 2005 J. Math. Phys. 46 033303) and the other one is the cycle mean field model. Both models show critical behaviour despite their rate functions having unique minimiser. The main focus is then a model with negative counter term, the probabilistic version of the so-called Huang–Yang–Luttinger model (van den Berg et al 1988 Commun. Math. Phys. 118 61–85). Criticality in this model is the existence of a critical parameter for which two simultaneous minimiser exists. At criticality an order parameter is introduced as the double limits for the density of cycles with diverging length, and as such it extends recent work in (Adams and Dickson 2021 Ann. Henri Poincaré 22 1535–60).