Limit Shapes for Gibbs Partitions of Sets

Large deviations analysis for random combinatorial partitions with counter terms

“…The reference measure. Random combinatorial partitions arise in many areas of mathematics as number theory, combinatorics, probability and statistical mechanics, as illustrated in [Ver96] and further developed in recent work [FS18a,FS18b].…”

“…The probability weights for the reference process are a special class of multiplicative weights. General multiplicative weights have been introduced by Vershik [Ver96] and are studied and analysed for scaling limits in [FS18a,FS18b]. Our study is not concerned with scaling limits but with large deviation limits as N → ∞.…”

“…The name refers to the number of physical particles in the system. The measure, though well-known in the physics literature, is different in character from the partitions weights studied in [Ver96] and [FS18a,FS18b]. Namely, as the earlier weights are just product of the single weights, the squared term in the Hamiltonian H (P M F ) creates product of weights of pairwise different weight numbers.…”

“…In future work we shall address scaling limits as in [Ver96,FS18a,FS18b] where we hope that the detailed analysis of the rate function zeroes can help establishing different scaling limits. Another question concerns the concentration of measure around the two distant rate function zeroes of the HYL model at criticality.…”

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In this paper, we study various models for random combinatorial partitions using large deviation analysis for diverging scale of the reference process. Scaling limits of similar models have been studied recently [FS18a,FS18b] going back to [Ver96]. After studying the reference model, we provide a complete analysis of two mean field models, one of which is well-know [BCMP05] and the other one is the cycle mean field model.

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