Samuel Modée: Limiting Behavior of the Largest Eigenvalues of Random Toeplitz Matrices

Abstract

The subject of my talk is random symmetric Toeplitz matrices of size n. Assuming that the distinct entries are independent centered random variables with finite \(\gamma\)-th moment (\(\gamma>2\)), I will establish a law of large numbers for the largest eigenvalue. Following the approach of Sen and Virág (2013), in the large \(n\) limit, the largest eigenvalue rescaled by \(\sqrt{2n\, log(n)}\) is shown to converge to the limit \(0.8288...\) . I will explain the background theory and also present some related results. Finally, a numerical analysis is used to illustrate the rate of convergence and the oscillatory nature of the eigenvectors of random Toeplitz matrices.