Ph.D Student | Erlihson Michael |
---|---|

Subject | Asymptotics of Multiplicative Measures and Transient Behavior of Coagulation-Fragmentation Processes |

Department | Department of Applied Mathematics |

Supervisor | Professor Boris Granovsky |

The thesis consists of three independent chapters.

In the first
chapter we study asymptotic properties of a class of multiplicative measures on
the set of partitions. We find limit shapes for a family of multiplicative
measures on the set of partitions, induced by the exponential generating
functions with expansive parameters, _{}, where *C *is a positive
constant. The measures considered are associated with reversible
coagulation-fragmentation processes and decomposable random combinatorial
structures, known as assemblies. We prove a functional central limit theorem
for fluctuations of a properly scaled integer partition chosen randomly
according to the above measure, from its limit shape, as the size *N *of a
structure goes to infinity. We reveal that numbers of components (occupation
numbers) of sizes are asymptotically independent, as_{}, while the occupation
numbers of sizes* _{}*become conditionally
independent given their masses. Thus,

In the second chapter we investigate the asymptotics of the number of decomposable combinatorial structures for all three basic types of such structures: assemblies, multisets and selections. Our study is devoted to two objectives:

*1. *Extension
of the seminal Meinardus(1954) theorem from weighted partitions to two other types
of structures, by providing a unified probabilistic approach for the
asymptotics of these combinatorial structures;

*2. *Weakening
of one of the three Meinardus conditions.

As a result, we obtain asymptotic formulae for numbers of multisets, selections and assemblies, under a new version of Meinardus sufficient conditions on parameters of combinatorial structures. We also discuss the striking similarity between the derived asymptotic formulae and provide clarifications of the context of the conditions of the theorem.

The third
chapter is devoted to the study of the transient behavior of
coagulation-fragmentation processes (CFP’s). We derive necessary and sufficient
conditions for time homogeneity of birth-death processes depicting the number
of clusters in a CFP at time *t *and provide a characterization of
solvable (in the sense of the expected number of clusters) CFP’s. We also show
that any solvable CFP exhibits gelation.