.Stat.M ech.(2009)P05012 ournal of Statistical Mechanics: An IOP and SISSA journalJ Theory and Experiment Asymptotics of eigenvalues and

Abstract. A Toeplitz matrix is one in which the matrix elements are constant along diagonals. The Fisher–Hartwig matrices are much-studied singular matrices in the Toeplitz family. The matrices are defined for all orders, N. They are parameterized by two constants, α and β. Their spectrum of eigenvalues has a simple asymptotic form in the limit as N goes to infinity. Here we study the structure of their eigenvalues and eigenvectors in this limiting case. We specialize to the case with real α and β and 0 < α < |β | < 1, where the behavior is particularly simple. The eigenvalues are labeled by an index l which varies from 0 to N − 1. An asymptotic analysis using Wiener–Hopf methods indicates that for large N, the jth component of the lth eigenvector varies roughly in the fashion lnψlj ≈ iplj + O(1/N). The lth wavevector, pl, varies as pl = 2πl/N + i(2α + 1) lnN/N + O(1/N) (I) for negative values of β and values of l/(N − 1) not too close to zero or one. Correspondingly the lth eigenvalue is given b