On determinants identity minus Hankel matrix

TL;DR

This paper analyzes the asymptotic behavior of determinants of the form det(I_N - βH_N) for large N, where H_N is a Hankel matrix with jump discontinuities, revealing power-like asymptotics influenced by jump heights.

Contribution

It provides the first-order asymptotics of determinants involving Hankel matrices with discontinuous symbols, including explicit formulas for the Hilbert matrix case.

Findings

Determinants exhibit power-like asymptotics with exponents depending on jump heights.

Explicit asymptotic formula derived for the Hilbert matrix case.

Asymptotics involve arcsin functions of β, showing dependence on matrix discontinuities.

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 $\|H\|\leq 1$. 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 $$ \log \det(I_N - \beta \mathbf{H}_N) = -\frac{\log N}{2\pi^2} \big(\pi\arcsin(\beta)+\arcsin^2(\beta)+o(1)\big),\qquad N\to\infty. $$