On determinants identity minus Hankel matrix

Abstract

In this note, we study the asymptotics of the determinant $\det(I_N - βH_N)$ 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 $\|H\|\leq 1$. Moreover, we assume $β\in\mathbb C$ with $|β|<1$ 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 $π^{-1}(j+k+1)^{-1}$, where $j,k\in \mathbb Z_+$ we obtain $$ \log \det(I_N - β\mathbf{H}_N) = -\frac{\log N}{2π^2} \big(π\arcsin(β)+\arcsin^2(β)+o(1)\big),\qquad N\to\infty. $$