Some Universality Results Of Random Matrix Theory

In this thesis, we will study some universal properties of Random matrices. The word "universal properties" means some properties which are only related to the structures of the random matrices but not the details of the distributions of the matrix elements. Four topics will be included in this thesis. Specifically, we will discuss the limiting distributions of the extreme eigenvalues of sample correlation matrices, Gaussian fluctuations of partial linear eigenvalue statistics of Wigner ma-trices, logarithmic law of random determinant and a global property of eigenvectors of Wigner matrices.Chapter1will be devoted to the introduction of some backgrounds of these problems and some preliminaries.In Chapter2. we will study the extreme eigenvalues of sample correlation ma-trices. Let the sample correlation matrix be W=YYT, where Y=(yij)p,n with yij=xij/(?).We assume{xij:1≤i≤p,1≤j≤n} to be a collection of independent symmetrically distributed random variables with sub-exponential tails. Moreover, for any i, we assume xij,1≤j≤n to be identically distributed. We assume0<p<n and p/nâ†'y with some y∈(0,1) as p,nâ†'∞.In this paper, we provide the Tracy-Widom law (TW1) for both the largest and smallest eigenvalues of W. If xij are i.i.d. standard normal, we can derive the TW1for both the largest and smallest eigenvalues of the matrix R=RRT, where R=(rij)p,n withIn Chapter3,we study the complex Wigner matrices Mn=1/(?)Wn whose eigenvalues are typically in the interval [-2,2]. Let λ1≤λ2…≤λn be the ordered eigenvalues of Mn. Under the assumption of four matching moments with the Gaus-sian Unitary Ensemble(GUE), for test function f4-times continuously differentiable on an open interval including [-2,2], we establish central limit theorems for two types of partial linear statistics of the eigenvalues. The first type is defined with a threshold u in the bulk of the Wigner semicircle law as An[f;u]=∑l=1n f(λl)1{λl≤u}. And the second one is Bn[f;k]=∑l=1k f(λl) with positive integer k=kn such that k/nâ†'y E (0,1) as n tends to infinity. Moreover, we; derive; a weak convergence result for a partial sum process constructed from Bn[f;(?)nt(?)]. The; main difficulty is to deal with the linear eigenvalue statistics for the test functions with several non-differentiable points. And our main strategy is to combine the Helffer-Sjostrand formula and a comparison procedure on the resolvents to extend the results from GUE case to general Wigner matrices case. Moreover, the results on An[f;u] for the real Wigner matrices will also be briefly discussed.In Chapter4, we consider the square random matrix An=(aij)n,n, where {dij=:aij(n),i,j=1,…,n} is a collection of independent real random variables with means zero and variances one. Under the additional moment condition we prove Girko’s logarithmic law of det An in the sense that as nâ†'∞Chapter5will be devoted to studying a global property of eigenvectors of Wigner matrices. Let Mn be an n×n real (resp. complex) Wigner matrix and Un∧nUn*be its spectral decomposition. Set (y1,y2…,yn)T=Un*x, where x=(x1,x2,…,xn)T is a real (resp. complex) unit vector. Under the assumption that the elements of Mn have4matching moments with those of GOE (resp. GUE), we show that the process converges weakly to the Brownian bridge for any x satisfying‖x‖∞â†'0as nâ†'∞, where β=1for the real case and β=2for the complex case. Such a result indicates that the othorgonal (resp. unitary) matrices with columns being the eigenvectors of Wigner matrices are asymptotically Haar distributed on the orthorgonal (resp. unitary) group from a certain perspective.At last, we will present a short summary and make a brief plan for future work.