On determinants identity minus Hankel matrix

Abstract

In this note, we study the asymptotics of the determinant $\det (I_N-\beta H_N)$ for $N$ large, where $H_N$ is the $N\times N$ restriction of a Hankel matrix $H$ with finitely many jump discontinuities in its symbol satisfying $\Vert H\Vert \le 1$. Moreover, we assume $\beta \in \mathbb{C}$ with $|\beta |<1$ and $I_N$ denotes the identity matrix. We determine the first order asymtoptics as $N\to \infty$ of such determinants and show that they exhibit power-like asymptotic behaviour, with exponent depending on the height of the jumps. For example, for the $N\times N$ truncation of the Hilbert matrix $\mathbf{H}$ with matrix elements ${\pi }^{-1}(j+k+1{)}^{-1}$, where $j,k\in {\mathbb{Z}}_{+}$ we obtain $$\log \det (I_N-\beta {\mathbf{H}}_N)=-\frac{\log N}{2{\pi }^2}(\pi \arcsin (\beta )+{\arcsin }^2(\beta )+o(1)),N\to \infty .$$