On Toeplitz determinants with slow Fourier decay

Abstract

We study Toeplitz determinants $\det T_n(e^f)$ for $f$ whose Fourier coefficients satisfy $f_k=O(|k|^{-1})$. This regime extends beyond $H^{1/2}$ and includes symbols with Fisher-Hartwig singularities. We develop an operator-theoretic approach based on the Baker-Campbell-Hausdorff formula that separates the quadratic term \[ \sum_{k=1}^{\infty}\min(k,n)f_kf_{-k} \] from the higher-order terms in the expansion of $\log\det T_n(e^{tf})$. We show that this quadratic term accounts for the possible growth with $n$, while every fixed higher-order coefficient remains bounded. For symbols with bounded positive and negative Fourier parts, our estimates yield two-sided bounds for the determinant after removal of the quadratic contribution. For a broader admissible class, including Fisher-Hartwig-type symbols, we obtain uniform higher-order coefficient bounds and a central limit theorem for the associated CUE linear statistics. We also obtain bounds on mixed exponential moments for CUE-derived random fields beyond the characteristic polynomial.