## Date of Submission

9-28-2010

## Date of Award

9-28-2011

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Doctoral Thesis

## Degree Name

Doctor of Philosophy

## Subject Name

Mathematics

## Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

## Supervisor

Bose, Arup (TSMU-Kolkata; ISI)

## Abstract (Summary of the Work)

Consider a sequence of matrices whose dimension increases to infinity. Suppose the entries of this sequence of matrices are random. These matrices with increasing dimension are called large dimensional random matrices (LDRM).Practices of random matrices, more precisely the properties of their eigenvalues, has emerged first from data analysis (beginning with Wishart (1928) [132]) and then from statistical models for heavy nuclei atoms (beginning with Wigner (1955) [130]). To insist on its physical applications, a mathematical theory of the spectrum of the random matrices began to emerge with the work of E. P. Wigner, F. J. Dyson, M. L. Mehta, C. E. Porter and co-workers in the 1960â€™s. And this established the link between various branches of mathematics including classical analysis and number theory. Slowly it appeared in other branches of sciences as well, like high dimensional data analysis, communication theory, dynamical systems, finance, diffusion process and so on. The most important papers on random matrix theory in physics from this early period are collected in the book edited by Porter (1965) [102].Initially enumerative combinatorics was the only, though very useful, tool to analyze random matrices. Many other sophisticated and varied mathematical tools are now available in the field. These includes Fredholm determinants (in the 1960â€™s), diffusion processes (in the 1960â€™s), integrable systems (in the 1980â€™s and early 1990â€™s), and the theory of free probability (in the 1990â€™s). Many of the mathematical elements of random matrix theory which were developed in the beginning of the 1960â€™s has been described in the book by Mehta (2004) [90].One of the most important objects to study in random matrix theory is the spectra of LDRM. The necessity of studying the spectra of LDRM, especially the Wigner matrices, arose in nuclear physics during the 1950â€™s. In quantum mechanics, the energy levels of quanta are not directly observable, but can be characterized by the eigenvalues of a matrix of observations. However the empirical spectral distribution (ESD) of theeigenvalues of a random matrix has a very complicated form when the order of the matrix is high. Many conjectures, e.g., the famous circular law conjecture were made through numerical computation.The random matrix literature is vast and evergrowing. We provide a very brief introduction restricting ourselves to areas/results which have some relevance to the problems considered in this thesis. For detailed information on these and for other developments we refer to the excellent books by Mehta (2004) [90], Bai and Silverstein (2010) [13], Anderson, Guionnet and Zeitouni (2010) [3] and the survey papers of Bai (1999) [10], Bose, Hazra and Saha (2010) [39].In Sections 0.1 and 0.2 we provide a brief summary of existing results on the limiting spectral distribution and on the extremes of eigenvalues. In this thesis we study the circulant and related random matrices. In Section 0.3 we provide some motivation to study such matrices. In Section 0.4 we give a brief summary of the thesis.

## Control Number

ISILib-TH304

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

## DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

## Recommended Citation

Saha, Koushik Dr., "Spectral Properties of Large Dimensional Random Circulant Type Matrices." (2011). *Doctoral Theses*. 26.

https://digitalcommons.isical.ac.in/doctoral-theses/26

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842802