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

TL;DR

This paper generalizes Szeg"o limit theorems for large matrices with decaying elements, using phase-space methods and Moyal star product expansions to derive explicit asymptotic determinant formulas.

Contribution

It introduces a systematic semiclassical expansion approach for asymptotic determinants of non-Toeplitz matrices with slowly varying elements.

Findings

Derived explicit asymptotic formulas for determinants.

Extended Szeg"o limit theorems beyond Toeplitz matrices.

Applied phase-space quantum mechanics techniques to matrix analysis.

Abstract

We work out a generalization of the Szeg\"o 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.