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Asymptotic behaviour of determinants through the expansion of the Moyal star product

Maurizio Fagotti, Vanja Marić

TL;DR

The paper develops a comprehensive framework to analyze the asymptotic determinants of large matrices that are not strictly Toeplitz but have exponentially decaying off-diagonal elements. It combines a phase-space viewpoint via the $\,Moyal\,$ star product with Wiener–Hopf factorizations to extend Szegö-type determinant theorems to star-Toeplitz matrices, including slow-diagonal variations and block structures. The main contributions are a Borodin–Okounkov–Case–Geronimo type determinant formula, Szegö-type limit theorems, and an explicit semiclassical expansion for locally Toeplitz and locally Toeplitz-flow matrices, with detailed treatment of corner and bulk terms and regularisation. These results yield explicit asymptotic expressions for determinants in inhomogeneous systems, with potential applications to correlation functions and entanglement entropies in quantum many-body setups, while highlighting the need for symbol regularisation in the presence of singularities or nonzero star winding numbers.

Abstract

We work out a generalization of the Szegö limit theorems on the determinant of large matrices. We focus on matrices with nonzero leading principal minors and elements that decay to zero exponentially fast with the distance from the main diagonal, but we relax the constraint of the Toeplitz structure. We obtain an expression for the asymptotic behaviour of the determinant written in terms of the factors of a left and right Wiener-Hopf type factorization of an appropriately defined symbol. For matrices with elements varying slowly along the diagonals (e.g., in locally Toeplitz sequences), we propose to apply the analogue of the semiclassical expansion of the Moyal star product in phase-space quantum mechanics. This is a systematic method that provides approximations up to any order in the typical scale of the inhomogeneity and allows us to obtain explicit asymptotic formulas.

Asymptotic behaviour of determinants through the expansion of the Moyal star product

TL;DR

The paper develops a comprehensive framework to analyze the asymptotic determinants of large matrices that are not strictly Toeplitz but have exponentially decaying off-diagonal elements. It combines a phase-space viewpoint via the star product with Wiener–Hopf factorizations to extend Szegö-type determinant theorems to star-Toeplitz matrices, including slow-diagonal variations and block structures. The main contributions are a Borodin–Okounkov–Case–Geronimo type determinant formula, Szegö-type limit theorems, and an explicit semiclassical expansion for locally Toeplitz and locally Toeplitz-flow matrices, with detailed treatment of corner and bulk terms and regularisation. These results yield explicit asymptotic expressions for determinants in inhomogeneous systems, with potential applications to correlation functions and entanglement entropies in quantum many-body setups, while highlighting the need for symbol regularisation in the presence of singularities or nonzero star winding numbers.

Abstract

We work out a generalization of the Szegö limit theorems on the determinant of large matrices. We focus on matrices with nonzero leading principal minors and elements that decay to zero exponentially fast with the distance from the main diagonal, but we relax the constraint of the Toeplitz structure. We obtain an expression for the asymptotic behaviour of the determinant written in terms of the factors of a left and right Wiener-Hopf type factorization of an appropriately defined symbol. For matrices with elements varying slowly along the diagonals (e.g., in locally Toeplitz sequences), we propose to apply the analogue of the semiclassical expansion of the Moyal star product in phase-space quantum mechanics. This is a systematic method that provides approximations up to any order in the typical scale of the inhomogeneity and allows us to obtain explicit asymptotic formulas.
Paper Structure (23 sections, 33 theorems, 182 equations, 1 table)

This paper contains 23 sections, 33 theorems, 182 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notations
  3. Overview of the Szegö limit theorems
  4. Statement of the problem and organization of the article
  5. Main results
  6. Readaptation of standard results
  7. Asymptotic expansion
  8. On block matrices
  9. Examples
  10. On the Wiener-Hopf star factorization.
  11. On the asymptotic expansion.
  12. On block-Toeplitz matrices.
  13. Limit behavior of determinants
  14. A Borodin-Okounkov-Case-Geronimo type formula---proof of Proposition \ref{['t:BOCG']}
  15. Szegö type limit theorems---proof of theorem \ref{['c:strong']}
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $a\in L^{\infty}(\mathbb T)$ satisfy Then

Theorems & Definitions (109)

  • Theorem 1.1: Strong Szegö Limit Theorem
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Remark 2.1
  • Definition 2.3: Symbol
  • Remark 2.2
  • Definition 2.4: Moyal-Weyl–Groenewold product
  • Remark 2.3
  • Definition 2.5: Star inverse
  • ...and 99 more