The first order asymptotics of the extreme eigenvectors of certain Hermitian Toeplitz matrices (Q1030498)

Let \(T_{n}(a)\) be the \(n\times n\) Toeplitz matrix generated by a real-valued function \(a\in L^{1}(-\pi,\pi)\). Let \(\lambda_{1}^{(n)}\leq\cdots\leq\lambda_{n}^{(n)}\) denote the eigenvalues of this (Hermitian) matrix. The values \(\lambda^{(n)}_{k}\) for fixed \(k\) are called the extreme eigenvalues. Under some assumptions, the authors find the third order asymptotics of the extreme eigenvalues and the first order asymptotics of the corresponding eigenvectors as \(n\rightarrow \infty \). They also investigate some special cases arising in probability theory and physics.