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On Toeplitz determinants with slow Fourier decay

This paper develops an operator-theoretic approach using the Baker-Campbell-Hausdorff formula to analyze Toeplitz determinants with symbols exhibiting slow Fourier decay ( $f_k=O(|k|^{-1})$ ), demonstrating that their asymptotic behavior is governed by a quadratic term while higher-order coefficients remain bounded, thereby establishing two-sided bounds and a central limit theorem for associated Circular Unitary Ensemble linear statistics.

Original authors: Nedialko Bradinoff, Maurice Duits

Published 2026-08-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nedialko Bradinoff, Maurice Duits

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are listening to a complex piece of music, but instead of hearing the melody, you are trying to understand the hidden structure of the sound waves themselves. In the world of mathematics, specifically a field called "harmonic analysis," scientists study how complex patterns can be broken down into simple, rhythmic building blocks called Fourier coefficients. Think of these coefficients as the individual notes in a chord. Usually, when these notes get quieter and quieter as you go higher in pitch (a "fast decay"), the math is predictable and well-behaved. However, life gets messy when the notes don't fade away quickly; they linger, creating a "slow decay." This happens in real-world phenomena ranging from the behavior of particles in quantum physics to the mysterious zeros of the Riemann zeta function, which holds the key to understanding prime numbers. When these lingering notes interact, they create a mathematical puzzle known as a Toeplitz determinant. For decades, mathematicians could only solve this puzzle for smooth, well-behaved sounds. But what happens when the sound is rough, jagged, or even has sudden "screeches" (singularities)? That is the question this paper tackles.

The authors, Nedialko Bradinoff and Maurice Duits, have developed a new way to measure these messy, slow-decaying sounds. They focus on a specific type of mathematical object called a Toeplitz determinant, which acts like a giant calculator for the collective behavior of random systems, such as the energy levels in a quantum gas or the eigenvalues of a random matrix. The paper's main finding is that even when the underlying "notes" (Fourier coefficients) decay very slowly—specifically, when they drop off at a rate of $1/|k|$ —the system still follows a surprisingly orderly pattern. The authors prove that the chaotic growth of the determinant is almost entirely driven by a single, simple "quadratic" term (a specific type of interaction between the notes). Once you account for this main driver, all the remaining, more complex interactions (the higher-order terms) stay surprisingly small and bounded, rather than exploding out of control.

To reach this conclusion, the researchers used a clever mathematical tool called the Baker–Campbell–Hausdorff formula. You can think of this as a sophisticated recipe for mixing ingredients. If you have two different ingredients (operators) and you mix them in a specific order, the result isn't just a simple sum; it's a complex interaction. This formula allows the authors to separate the "big, loud" interactions from the "quiet, subtle" ones. They showed that for a broad class of symbols—including those with the famous Fisher–Hartwig singularities, which look like sharp spikes in the data—the "loud" part is exactly what causes the determinant to grow with the size of the system ( $n$ ). The "quiet" parts, which represent the complex, higher-order chaos, remain under control. This is a significant shift in perspective because, for a long time, mathematicians thought the wild growth was caused by the specific shape of the singularities (the spikes). This paper argues that the growth is actually caused by the slowness of the decay, regardless of whether the spikes are there or not.

The paper provides rigorous proofs for these claims, establishing strict upper and lower bounds for these determinants. They demonstrate that for a wide range of functions, including those with unbounded singularities, the "remainder" of the calculation (everything after the main quadratic term) is bounded by constants that do not depend on the size of the system. This means that even as the system gets infinitely large, the chaotic noise doesn't get worse; it just stays within a predictable range. The authors also connect this to the Circular Unitary Ensemble (CUE), a model for random unitary matrices used in quantum physics. They show that the linear statistics of these matrices (essentially, summing up the values of a function over the matrix's eigenvalues) follow a Central Limit Theorem, meaning they behave like a normal bell curve, even when the test functions have these slow-decaying, singular characteristics.

In simpler terms, the authors have built a new lens that lets us see order in what looked like chaos. They proved that the "noise" in these complex systems is actually quite tame once you subtract the main, predictable trend. This is particularly exciting for applications involving the characteristic polynomial of random matrices, which is linked to the distribution of prime numbers and the behavior of log-correlated fields. The paper suggests that the "freezing transition" seen in these fields—where the system suddenly changes behavior—is driven by the slow decay of the Fourier coefficients, not just by the presence of singularities. While the authors admit their bounds might not be the absolute tightest possible (they are "crude" in some estimates), they are robust and apply to a much wider class of functions than previous methods. They have effectively shown that the universe of these mathematical determinants is more uniform and predictable than we previously thought, provided we know how to separate the signal from the noise.

Technical Summary: On Toeplitz Determinants with Slow Fourier Decay

Problem Statement
The paper investigates the asymptotic behavior of Toeplitz determinants $D_n(a) = \det T_n(a)$ as the matrix size $n \to \infty$ , specifically for symbols $a = e^f$ where the Fourier coefficients of the logarithm $f$ exhibit slow decay. The primary regime of interest is defined by the condition $|(f)_k| = O(|k|^{-1})$ as $|k| \to \infty$ .

This condition places the problem just beyond the scope of the classical Strong Szegő Limit Theorem, which typically requires $f$ to belong to the Sobolev space $H^{1/2}$ (i.e., $\sum k |f_k|^2 < \infty$ ). The slow decay regime includes the Fisher–Hartwig class of symbols, which feature logarithmic singularities and jump discontinuities. In the Fisher–Hartwig setting, the determinant asymptotics include an additional power of $n$ (specifically $n^{\sum (\alpha_j^2 - \beta_j^2)}$ ) compared to the standard Szegő limit. The authors aim to determine whether this additional growth is a universal feature of slow Fourier decay or a specific artifact of the local singularity structure, and to provide uniform estimates that extend beyond the Fisher–Hartwig class.

Methodology
The authors develop a new operator-theoretic approach based on the Baker–Campbell–Hausdorff (BCH) formula. The methodology proceeds through three main stages:

Regularization and Factorization: The Toeplitz determinant is represented as the determinant of a finite section of a product of exponentials of Toeplitz operators. Using the Wiener–Hopf factorization $e^f = e^{f_-}e^{f_+}$ (where $f_+$ is analytic inside the unit disk and $f_-$ outside), the determinant is rewritten to isolate a multiplicative commutator:
$\det T_n(e^{tf}) = \exp\left( nf_0 + \sum_{k=1}^\infty \min(k, n) f_k f_{-k} \right) \det(P_n e^{Z(t)} P_n + Q_n)$
Here, $Z(t)$ arises from the commutator $e^{-tT(f_+)}e^{tT(f_-)}e^{tT(f_+)}e^{-tT(f_-)}$ .
BCH Expansion: The multiplicative commutator is expanded using the BCH formula into a single exponential $e^{Z(t)}$ , where $Z(t)$ is a Lie series involving nested commutators of the Toeplitz operators $T(f_+)$ and $T(f_-)$ .
$Z(t) = -t^2 [T(f_+), T(f_-)] + \frac{t^3}{2} [T(f_+) - T(f_-), [T(f_+), T(f_-)]] + \dots$
The quadratic term corresponds to the known growth in the Fisher–Hartwig case. The higher-order terms ( $m \ge 3$ ) represent the remainder.
Norm Estimates and Determinant Bounds: The core technical contribution involves bounding the norms of these nested commutators. The authors distinguish between two cases:
- Bounded Symbols: Where the positive and negative Fourier parts ( $f_+$ and $f_-$ ) are essentially bounded.
- Unbounded (Admissible) Symbols: Where $f_+$ or $f_-$ may be unbounded (e.g., Fisher–Hartwig singularities), but satisfy specific "admissibility" conditions involving bounded auxiliary functions.
The authors establish bounds on the operator norms and Hilbert–Schmidt norms of the "corners" (off-diagonal blocks) of these commutators. They utilize a general determinant inequality (from Duits and Johansson) which relates the logarithm of the determinant of a finite section to the trace of the operator, provided the off-diagonal corners are Hilbert–Schmidt with controlled norms.

Key Contributions and Results

Separation of Growth: The paper demonstrates that the additional power of $n$ observed in Fisher–Hartwig asymptotics is entirely contained within the quadratic term $\sum_{k=1}^\infty \min(k, n) f_k f_{-k}$ of the expansion. The higher-order cumulants (coefficients of $t^m$ for $m \ge 3$ in $\log \det T_n(e^{tf})$ ) remain bounded as $n \to \infty$ , even when the variance grows logarithmically.
Uniform Bounds for Bounded Symbols (Theorem 2.1): For symbols where $f_\pm$ are bounded and decay as $O(|k|^{-1})$ , the authors prove that the $m$ -th cumulant $C_m^{(n)}(f)$ satisfies $|C_m^{(n)}(f)| \le a c^m$ for constants independent of $n$ . This implies that the remainder term in the determinant expansion is uniformly bounded.
Bounds for Unbounded/Admissible Symbols (Theorem 2.10): For a broader class of "admissible" symbols (including Fisher–Hartwig types), the authors derive bounds on the cumulants that grow super-exponentially in $m$ (specifically $O((cm)^m)$ ) but remain uniform in $n$ . While this prevents summing the series to a full asymptotic expansion, it is sufficient to control the remainder for fixed $m$ .
Central Limit Theorem (Corollary 2.12): The results imply a Central Limit Theorem for linear statistics of the Circular Unitary Ensemble (CUE) for admissible symbols. Specifically, $X_n(f) - \mathbb{E}X_n(f)$ , normalized by the square root of the variance (which grows with $n$ ), converges in distribution to a standard normal variable. This extends known results to cases where $f \notin H^{1/2}$ .
Mixed Exponential Moments (Corollary 2.5): For bounded symbols, the paper provides two-sided bounds on mixed exponential moments of linear statistics. These bounds are uniform in the locations of the evaluation points, a feature crucial for analyzing log-correlated fields and characteristic polynomials where singularities may merge.

Significance and Claims
The authors claim that their approach shifts the understanding of Toeplitz determinant asymptotics from a focus on the local nature of singularities (Fisher–Hartwig) to the global decay rate of Fourier coefficients. By isolating the quadratic contribution, they show that the "extra" growth in the determinant is a consequence of slow Fourier decay rather than the specific form of the singularity.

The methodology offers greater robustness and generality than previous Riemann-Hilbert approaches, particularly in handling regimes where singularities merge (e.g., at distances of order $n^{-\delta}$ ), a scenario where classical Fisher–Hartwig asymptotics become ill-suited. While the authors acknowledge that their bounds for unbounded symbols are not optimal (growing super-exponentially in the order of the cumulant), they assert that these estimates are sufficient to establish the Central Limit Theorem and provide uniform control over mixed moments, which are key ingredients in the study of Gaussian Multiplicative Chaos and the extreme values of the Riemann zeta function.

The paper does not claim to provide a full asymptotic expansion with explicit constant terms for the general unbounded case, noting that this comes at the expense of the explicit constant term in exchange for uniformity in parameters. The results are presented as a step toward understanding the universality of log-correlated field behaviors in random matrix theory beyond the characteristic polynomial.

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