Abstracts Mathematics

On the stability of random matrices

by Nikolaos Poulios

The aim of this thesis is multiple. Initially we study the diffusion process on dynamicalfinancial networks. To be more precise, we study the effect of diffusion method to interbanknetworks in relation to connected, directed and weighted networks. We consider networks ofn different banks which exchange funds (loans) and the main feature is how the leverage'sof banks can be chosen to improve the financial stability of the network. This is done byconsidering differential equations of diffusion type.Furthermore, we investigate the problem of almost zeros of polynomial matrices as usedin the system theory. It is related to the controllability and observability notion of systems aswell as the determination of Macmillan degree and complexity of systems. We also presentsome new results in this important invariant in the light of randomness and we prove anuncertainty type relation appearing in such ensembles of operators.In addition, we introduce the concept of ε-uncontrollability for random linear systems,i.e. a linear system in which the usual matrices have been replaced by random matrices. Wealso estimate the ε-uncontrollability in the case where the matrices come from the Gaussianorthogonal ensemble. Our proof utilizes tools from systems theory, probability theory andconvex geometry.This thesis is divided into three parts: Introduction and literature review material in PartI, methodological tools which were used in this thesis in Part II, and the rest consists of threeresearch papers among which the first one has been published in a collected volume underSpinger publications and the other two have been published in Institute of Mathematics andits Applications journal (IMA), in Part III.In Part I and especially in Chapter 1 we present an introduction and we provide information related to the motivation and the structure of this thesis. In addition in Chapter 2 we havea literature review about what other scientists have done so far on topics like mathematicalcontrol theory, diffusion process, graphs/networks and random matrix theory.The methodological tools needed for the derivation of the main research results presentedin Part III are given in Part II, Chapter 3. Furthermore, in this part, topics and notions frommathematical control theory, like controllability/observability, are presented. Moreover, wedescribe not only the meaning of RMT but also the the given examples of what a randommatrix is and what makes it so special. Another topic that is covered in this Chapter is graphtheory and examples about connected, directed and weighted graphs/networks and algebraicproperties of graphs are given. Finally, the topic of diffusion process is presented as well.Part III constitutes the most important part of this research study which is presented inthis thesis. Thus, in Chapter 4, we describe the diffusion process on connected, directed andweighted interbank loan networks. We prove that diffusion can drive the bank network to itssteady state where the leverages of all the banks of the network are equal. Diffusion in thebank…