Asymptotic formula for a partition function of reversible coagulation-fragmentation processes

Abstract

We construct a probability model seemingly unrelated to the considered stochastic process of coagulation and fragmentation. By proving for this model the local limit theorem, we establish the asymptotic formula for the partition function of the equilibrium measure for a wide class of parameter functions of the process. This formula proves the conjecture stated in [5] for the above class of processes. The method used goes back to A.Khintchine.

Clustering in coagulation-fragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and limiting laws

“…Our subsequent asymptotic analysis extends the one in [20] in two different directions: from c n to c…”

“…IV, V (see also [20]). Independently of the context of the problem considered, the implementation of Khintchine's method for deriving asymptotic formulae always follows the following two-step scheme:…”

“…was established in [20], with the help of Khintchine's method. Our primary aim in this section will be to extend the method to the aforementioned problem (2.14).…”

“…The probabilistic model suggested below is a modification of the one in [20]. We start by setting in (2.12), (2.13)…”

“…As examples, see (in chronological order) [19] of G. Freiman, [36], Ch. 2 of A. Postnikov, [21] of G. Freiman and J. Pitman, [33] of R. Mutafchiev, [31] of V. Kolchin, [20] of G. Freiman and B. Granovsky, and [22] of G. Freiman, A. Vershik and Yu. Yakubovitz.…”