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 $δ\in \mathbb{R} \setminus \{-1, -2, \ldots\}$, 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 $δ$ 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 $θ_δ:= δ$ if $δ< -1/2$, and $θ_δ:= \frac 1 π\arcsin(\sin(δπ))$ if $δ\ge -1/2$ and $\arcsin \colon [-1,1] \to [-\fracπ{2}, \fracπ{2}]$ denotes the principal branch of $\arcsin$. For $δ\geq -\frac 12$ the asymptotics is known. The novelty of the present paper is the regime $δ< - \frac 12$ and understanding the dichotomy in the decay exponent. This stems from the $\pm 1$ edge eigenvalues of the limiting operator appearing for $δ< - \frac 12$. Most notably, we obtain for $δ< -\frac12$ \[ \log \det\bigl(I_N - (H_N^δ)^2\bigr) = -δ^2 \log N + O(1). \] Such determinants arise in the study of Anderson&#39;s orthogonality catastrophe.