The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Physical Review B

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Methods Of Modern Mathematical Physics

We consider the ensemble of adjacency matrices of Erdős–Renyi random graphs, that is, graphs on N vertices where every edge is chosen independently and with probability p≡p(N). We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as pN→∞ (with a speed at least logarithmic in N), the density of eigenvalues of the Erdős–Renyi ensemble is given by the Wigner semicircle law for spectral windows of length larger than N−1 (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the l∞-norms of the l2-normalized eigenvectors are at most of order N−1/2 with a very high probability. The estimates in this paper will be used in the companion paper [Spectral statistics of Erdős–Renyi graphs II: Eigenvalue spacing and the extreme eigenvalues (2011) Preprint] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pN≫N2/3.

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Spectral statistics of Erdős–Rényi graphs I: Local semicircle law

Motivated by the eigenvalue problems for ordinary and partial differential operators, we shall develop the spectral theory for linear operators in Hilbert spaces. Here we transform the unbounded differential operators into singular integral operators which are completely continuous. With his study of integral equations D. Hilbert, together with his students E. Schmidt, I. Schur, and H. Weyl, opened a new era for the analysis.

Linear Operators in Hilbert Spaces

We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrice under the assumption that the off-diagonal matrix entries have uniformly bounded fifth moment and the diagonal entries have uniformly bounded third moment. Using our recent results on the fluctuation of resolvent entries [31],[28], and ideas from [9], we extend results by M.Capitaine, C.Donati-Martin, and D.F\'eral [12], [13].

On Finite Rank Deformations of Wigner Matrices

Nous etudions la distribution des valeurs propres qui sortent de l’amas du spectre de matrices de Wigner deformees par une matrice de rang fini sous l’hypothese que les valeurs absolues des elements non diagonaux aient un moment d’ordre cinq uniformement borne et que valeurs absolues des elements diagonaux aient un moment d’ordre trois uniformement borne. En utilisant des travaux recents (On fluctuations of matrix entries of regular functions of Wigner matrices with non-identically distributed entries, Unpublished manuscript; Fluctuations of matrix entries of regular functions of Wigner matrices, Unpublished manuscript) et des idees de (Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices, Unpublished manuscript), nous etendons les resultats de Capitaine, Donati-Martin et Feral (Ann. Probab. 37 (2009) 1–47; Ann. Inst. Henri Poincare Probab. Stat. 48 (2012) 107–133).

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On finite rank deformations of Wigner matrices

For fixed $$m > 1$$
 , we study the product of $$m$$
 independent $$N \times N$$
 elliptic random matrices as $$N$$
 tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability $$1$$
 , to the $$m$$
 -th power of the circular law, regardless of the joint distribution of the mirror entries in each matrix. This leads to a new kind of universality phenomenon: the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves. Our result also generalizes earlier results of Gotze–Tikhomirov (On the asymptotic spectrum of products of independent random matrices, available at http://arxiv.org/abs/1012.2710
 ) and O’Rourke–Soshnikov (J Probab 16(81):2219–2245, 2011) concerning the product of independent iid random matrices.

Products of Independent Elliptic Random Matrices

We study the asymptotic behavior of outliers in the spectrum of bounded rank perturbations of large random matrices. In particular, we consider perturbations of elliptic random matrices which generalize both Wigner random matrices and non-Hermitian random matrices with iid entries. As a consequence, we recover the results of Capitaine, Donati-Martin, and Feral for perturbed Wigner matrices as well as the results of Tao for perturbed random matrices with iid entries. Along the way, we prove a number of interesting results concerning elliptic random matrices whose entries have finite fourth moment; these results include a bound on the least singular value and the asymptotic behavior of the spectral radius.

Low rank perturbations of large elliptic random matrices

We consider a class of elliptic random matrices which generalize two classical ensembles from random matrix theory: Wigner matrices and random matrices with iid entries. In particular, we establish a central limit theorem for linear eigenvalue statistics of real elliptic random matrices under the assumption that the test functions are analytic. As a corollary, we extend the results of Rider and Silverstein (Ann Probab 34(6):2118–2143, 2006) to real iid random matrices.

David Renfrew

Papers

On Finite Rank Deformations of Wigner Matrices

On finite rank deformations of Wigner matrices

Products of Independent Elliptic Random Matrices

Low rank perturbations of large elliptic random matrices

Central Limit Theorem for Linear Eigenvalue Statistics of Elliptic Random Matrices