Graduate Thesis Or Dissertation
Spectral Properties of Products of Independent Random Matrices Public Deposited
In the field of random matrix theory, there are many matrix models one may choose to study. This thesis focuses on independent and identically distributed (iid) random matrices. Given a random variable ξ, we say that a matrix is an iid random matrix if each entry is an iid copy of ξ, and we call ξ the atom random variable. Given a positive constant integer m, consider m random variables ξ1,...,ξm, and create an independent n ✕ n iid random matrix for each of these random variables. The results presented in this thesis focus on the limiting behavior of the eigenvalues of the product of these m independent iid random matrices. Call this product Pn and note that Pn is also an n ✕ n random matrix, but the entries are no longer independent.
This thesis is comprised of two main results. First, we examine the locations of eigenvalues of matrix products as the size of the matrices tend to infinity. From the previous results by O'Rourke, Renfrew, Soshnikov, and Vu, we see that under certain moment assumptions on the atom random variables, as n tends to infinity, the empirical spectral measure of the eigenvalues of the rescaled product n-m/2Pn converges to a measure supported on a disk centered at the origin in the complex plane with radius depending on the atom random variables. In this work, we study the asymptotic location of eigenvalues which can fall outside of this disk. These outlying eigenvalues may be present when the random matrix product Pn is additively perturbed by a low rank, small norm deterministic matrix. We also consider multiplicative perturbations, and perturbations in any order. By studying these various perturbations, we characterize when a perturbed matrix product has outliers, and the asymptotic locations of these outlying eigenvalues.
The second result of this thesis studies the fluctuation of the eigenvalues of the rescaled product n-m/2Pn. In particular, we define a linear statistic and use this to study the spectrum, or the collection of eigenvalues, of n-m/2Pn. We see that the limiting distribution of the unnormalized linear statistic is a mean-zero Gaussian distribution with variance depending on the the linear statistic. The explicit variance formula is computed as well.
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