It is known that in various random matrix models, large perturbations create outlier eigenvalues which lie, asymptotically, in the complement of the support of the limiting spectral density. This thesis studies fluctuations of these outlier eigenvalues of iid matrices @math under bounded rank and bounded operator norm perturbations @math , namely the fluctuations @math . The perturbations @math that we consider belong to a large class, where we allow for arbitrary Jordan types and almost minimal assumptions on the left and right eigenvectors. We obtain the joint convergence of the normalized asymptotic fluctuations of the outlier eigenvalues in this setting with a unified approach.
In @cite_17 , the outlier eigenvalues of perturbations of the single ring model are studied and their locations and limiting fluctuations are obtained ( [Theorem 2.9] rochet ) for finite rank and finite operator norm perturbations of arbitrary Jordan type. Note that the special case of the Ginibre ensemble, which is an iid matrix, is contained in this model as well. Our approach to dealing with perturbations of various Jordan types is similar and relies on a deterministic perturbation result known as the Lidskii-Vishik-Lyusternik perturbation theorem (see @cite_15 , @cite_14 , @cite_1 and references therein) which we have reproduced in Appendix .
@cite_15 @cite_14 @cite_1 @cite_17
2951358417 2003341284 2026417557 1503311242
This text is about spiked models of non Hermitian random matrices. More specifically, we consider matrices of the type @math , where the rank of @math stays bounded as the dimension goes to infinity and where the matrix @math is a non Hermitian random matrix, satisfying an isotropy hypothesis: its distribution is invariant under the left and right actions of the unitary group. The macroscopic eigenvalue distribution of such matrices is governed by the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We first prove that if @math has some eigenvalues out of the maximal circle of the single ring, then @math has some eigenvalues (called outliers) in the neighborhood of those of @math , which is not the case for the eigenvalues of @math in the inner cycle of the single ring. Then, we study the fluctuations of the outliers of @math around the eigenvalues of @math and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such facts had already been noticed for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of @math ) and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in the Hermitian case, but only in the case of non Gaussian models, whereas spiked Gaussian matrices belong to our model and can have such correlated outliers). Our first result generalizes a previous result by Tao for matrices with i.i.d. entries, whereas the second one (about the fluctuations) is new. Recently, the stability of certain topological phases of matter under weak perturbations was proven. Here, we present a short, alternate proof of the same result. We consider models of topological quantum order for which the unperturbed Hamiltonian H 0 can be written as a sum of local pairwise commuting projectors on a D-dimensional lattice. We consider a perturbed Hamiltonian H = H 0 + V involving a generic perturbation V that can be written as a sum of short-range bounded-norm interactions. We prove that if the strength of V is below a constant threshold value then H has well-defined spectral bands originating from the low-lying eigenvalues of H 0. These bands are separated from the rest of the spectrum and from each other by a constant gap. The width of the band originating from the smallest eigenvalue of H 0 decays faster than any power of the lattice size. We consider the spectral problem for the random Schrodinger operator on the multidimensional lattice torus increasing to the whole of lattice, with an i.i.d. potential (Anderson Hamiltonian). We obtain the explicit almost sure asymptotic expansion formulas for the extreme eigenvalues and eigenfunctions in the intermediate rank case, provided the upper distributional tails of potential decay at infinity slower than the double exponential function. For the fractional-exponential tails (including Weibull’s and Gaussian distributions), extremal type limit theorems for eigenvalues are proved, and the strong influence of parameters of the model on a specification of normalizing constants is described. In the proof we use the finite-rank perturbation arguments based on the cluster expansion for resolvents. Some integrated models for ranking scientific publications together with authors and journals are presented and analyzed. The models rely on certain adjacency matrices obtained from the relationships between citations, authors and publications, which together give a suitable irreducible stochastic matrix whose Perron vector provides the ranking. Some perturbation theorems concerning the Perron vectors of nonnegative irreducible matrices are proved. These theoretical results provide a validation of the consistency and effectiveness of our models. Several examples are reported together with some results obtained on a real set of data.