Local Statistical Properties Of Eigenvalues Of Random Matrix Products

The topic of random matrix products first introduced by Furstenberg and Kesten.Such products have been applied in the studies of the Lyapunove exponent in the dynamic systems,the Schrodinger operator theory,the wireless communi-cation,image processing,and so on.To determine the global and local statistical properties of the eigenvalues and singular values of a matrix ensemble as the ma-trix size goes to infinity is the is the core problem in the study of product random matrices.Recently,the limits of the empirical spectral distributions(global properties)of products of random matrices were obtained through the techniques from the free probability theory.But these techniques can not be applied to give the local statistical properties of the eigenvalues,and thus it is desirable to study these local properties.More recently,the joint probability density functions of the complex eigenvalues of several random matrix products(products of independent induced complex Ginibre matrices and their inverses,truncated unitary matrices and general statistical isotropic matrices)have been explicitly derived as determinantal point processes in a series of papers.These open up possibility of asymptotic analysis for local eigenvalue statistics.In this thesis we study the local statistics of eigenvalues of eigenvalues.This thesis includes:Firstly,we do some asymptotic analysis of a kind of multivariate integrals with singularity by investigating the saddle point method method for such a type of in-tegrals.This type of integrals is appeared in the integral representations of the correlation kernels of products of independent random matrices.Thus,the asymp-totics can be used to investigated the local statistical properties of the eigenvalues of these random matrices.Secondly,the eigenvalues of the product of m independent complex Ginibre matrix form a determinantal process.The correaltion kernel is the product of the weight function and the partial sum.We study the asymptotics of weight functions by apply the saddle point method,and represent the partial sums by multivariate complex integrals with singularity.By applying the asymptotics of such type of integrals established above,we prove that the eigenvalue correlation functions have the same local properties as those of the single complex Ginibre matrix,both in the bulk and at the edge of the spectrum.Meanwhile,the products of m independent truncated unitary matrices,and also the mixed products of truncated unitary ma-trices and induced complex Ginibre matrices are investigated.The limit of mean empirical spectral distributions and the local statistics of eigenvalues are obtained.Finally,the techniques from the optimal transportation theory are used to study the order statistics of the moduli of the eigenvalues of products of mN independent polynomial matrix ensembles with derivative type of order N.Based on the layered structured of the moduli of the eigenvalues,a replacement principle is established from the optimal transportation theory.By applying the replacement principle the limiting distribution of the largest n-th modular is obtained whenever mN/N converges to some constant ? ?[0,?).And,we investigate the relation between distributions of the largest eigenvalues(the spectral radii)and the local eigenvalue statistics of correlation functions by studying the asymptotics of the largest eigen-values of the spiked complex Wishart matrices.