Limit shapes of multiplicative measures associated with coagulation-fragmentation processes and random combinatorial structures

We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, ak ∼ Ck, k → ∞, p > 0, where C is a positive constant. The measures considered are associated with reversible coagulationfragmentation processes and combinatorial structures, known as assemblies. We prove a functional 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. Among other things, the paper also discusses, in a general setting, the interplay between limit shape, threshold and gelation.