טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentErlihson Michael
SubjectAsymptotics of Multiplicative Measures and Transient
Behavior of Coagulation-Fragmentation Processes
DepartmentDepartment of Applied Mathematics
Supervisor Mr. Boris Granovsky


Abstract

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, which is the threshold for the maximal component size is also a critical value for the asymptotic independence of the numbers of groups. The aforementioned results were obtained using a far reaching generalization of Khitchine’s probabilistic method. We also discuss the linkage between the following three important concepts in statistical mechanics and combinatorics: threshold, gelation, and limit shape. As a by-product of this study, we establish the non-existence of limit shapes for some known models.

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.