The Central Limit Theorem for Linear Spectral Statistics of Submatrices of the Gaussian Wigner Random Matrice

The main topic addressed here is the joint distribution of spectra of submatrices M(1) and M(2) of large Gaussian Wigner matrices M. A multidimensional central limit theorem for linear statistics of the eigenvalues of submatrices will be proved with explicit formulas for the covariance that relate the spectra to a random surface model known as the Gaussian free field. The regularity assumption is that test functions belong to the Sobolev space H s, for s > 5/2.;The organization is as follows. Chapters 1 and 2 consist of an introduction to Wigner matrices and the central limit theorem in the random matrix theory. Chapter 3 is a discussion of the results which motivated this work, in addition to an introduction to the Gaussian free field. Chapter 4 contains the new results of the author, and chapter 5 is an appendix describing some technical tools.