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 ∈ ℕ and δ∈ ℝ ∖ {−1, −2, …}, let HNδ := ( (sin((1+δ)π))/(π(j+k+1+δ)) )j,k=0N−1 be the generalized Hilbert matrix. For δ not a half-integer, we prove the large-N power-law asymptotics of the determinant log det(IN ± HNδ) = −(θδ2 ± θδ)/2 log N + O(1) where θδ:= δ if δ< −1/2, and θδ:= 1/πarcsin(sin(δπ)) if δ≥ −1/2 and arcsin : [−1,1] → [−π/2, π/2] denotes the principal branch of arcsin. For δ≥ −1/2 the asymptotics is known. The novelty of the present paper is the regime δ< − 1/2 and understanding the dichotomy in the decay exponent. This stems from the ± 1 edge eigenvalues of the limiting operator appearing for δ< − 1/2. Most notably, we obtain for δ< −1/2 log det(IN − (HNδ)2) = −δ2 log N + O(1). Such determinants arise in the study of Anderson's orthogonality catastrophe.
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BibTeX
@misc{gebert2026szegőtype,
title = {{Szegő-type determinant asymptotics for generalized Hilbert matrices with edge eigenvalues}},
author = {Martin Gebert and Heinrich Küttler},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00255},
eprint = {2609.00255},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00255},
}
RIS
TY - UNPB
TI - Szegő-type determinant asymptotics for generalized Hilbert matrices with edge eigenvalues
AU - Martin Gebert
AU - Heinrich Küttler
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00255
UR - https://arxiv.org/abs/2609.00255
AB - We compute the large-N asymptotics of the determinant of the identity plus or minus a generalized Hilbert matrix. For N ∈ ℕ and δ∈ ℝ ∖ {−1, −2, …}, let H_N^δ := ( (sin((1+δ)π))/(π(j+k+1+δ)) )_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 det(I_N ± H_N^δ) = −(θ_δ^2 ± θ_δ)/2 log N + O(1) where θ_δ:= δ if δ< −1/2, and θ_δ:= 1/πarcsin(sin(δπ)) if δ≥ −1/2 and arcsin : [−1,1] → [−π/2, π/2] denotes the principal branch of arcsin. For δ≥ −1/2 the asymptotics is known. The novelty of the present paper is the regime δ< − 1/2 and understanding the dichotomy in the decay exponent. This stems from the ± 1 edge eigenvalues of the limiting operator appearing for δ< − 1/2. Most notably, we obtain for δ< −1/2 log det(I_N − (H_N^δ)^2) = −δ^2 log N + O(1). Such determinants arise in the study of Anderson's orthogonality catastrophe.
ER -