Szegő-type determinant asymptotics for generalized Hilbert matrices with edge eigenvalues

Abstract

We compute the large-$N$ asymptotics of the determinant of the identity plus or minus a generalized Hilbert matrix. For $N\in \mathbb{N}$ and $\delta \in \mathbb{R}\setminus \{-1,-2,\dots \}$, let \[ H_N^δ := \left( \frac{\sin((1+δ)π)}{π(j+k+1+δ)} \right)_{j,k=0}^{N-1} \] be the generalized Hilbert matrix. For $\delta$ not a half-integer, we prove the large-$N$ power-law asymptotics of the determinant \[ \log \text{det}(I_N \pm H_N^δ) = -\frac{θ_δ^2 \pm θ_δ}{2} \log N + O(1) \] where ${\theta }_{\delta }:=\delta$ if $\delta <-1/2$, and ${\theta }_{\delta }:=\frac{1}{\pi }\arcsin (\sin (\delta \pi ))$ if $\delta \ge -1/2$ and $\arcsin :[-1,1]\to [-\frac{\pi }{2},\frac{\pi }{2}]$ denotes the principal branch of $\arcsin$. For $\delta \ge -\frac{1}{2}$ the asymptotics is known. The novelty of the present paper is the regime $\delta <-\frac{1}{2}$ and understanding the dichotomy in the decay exponent. This stems from the $\pm 1$ edge eigenvalues of the limiting operator appearing for $\delta <-\frac{1}{2}$. Most notably, we obtain for $\delta <-\frac{1}{2}$ \[ \log \det\bigl(I_N - (H_N^δ)^2\bigr) = -δ^2 \log N + O(1). \] Such determinants arise in the study of Anderson's orthogonality catastrophe.