Limit shapes of Gibbs distributions on the set of integer partitions : the expansive case

Erlihson, Michael M.; Granovsky, Boris L.

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 5, pp. 915-945.

Nous trouvons des formes limites pour une famille de mesures multiplicatives sur l’ensemble des partitions, induites par des fonctions génératrices exponentielles avec des paramètres d’expansion $a_k\sim C{k}^{p-1},k\to \infty ,p>0$, où $C$ est une constante positive. Les mesures considérées sont associées aux modèles Maxwell-Boltzmann généralisés de la mécanique statistique, des processus de coagulation-fragmentation réversibles et des structures combinatoires connues sous le nom d’assemblées. Nous prouvons un théorème de limite centrale pour les fluctuations d’une partition qui est mise à l'échelle convenablement et choisie aléatoirement selon la mesure ci-dessus. Nous démontrons que, quand la taille des composantes dépasse la valeur seuil, l’indépendance des nombres de composants se transforme en leur indépendance conditionnelle. Entre autres, cet article traite, dans un cadre général, des relations entre la forme limite, le seuil et la congélation.

We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, $a_k\sim C{k}^{p-1},k\to \infty ,p>0$, where $C$ is a positive constant. The measures considered are associated with the generalized Maxwell-Boltzmann models in statistical mechanics, reversible coagulation-fragmentation processes and combinatorial structures, known as assemblies. We prove a central limit theorem for fluctuations of a properly scaled partition chosen randomly according to the above measure, from its limit shape. We demonstrate that when the component size passes beyond the threshold value, the independence of numbers of components transforms into their conditional independence (given their masses). Among other things, the paper also discusses, in a general setting, the interplay between limit shape, threshold and gelation.

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