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

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.

45 pages; v3 accepted in Communications in Mathematical Physics

References in corpus (10)

• Quantum Quench in the Transverse Field Ising Chain II: Stationary State Properties
• Universal parity effects in the entanglement entropy of XX chains with open boundary conditions
• Universal corrections to scaling for block entanglement in spin-1/2 XX chains
• Boundary effects on symmetry resolved entanglement
• Non-equilibrium 1D many-body problems and asymptotic properties of Toeplitz determinants
• Bulk-boundary correspondence and singularity-filling in long-range free-fermion chains
• Nonequilibrium symmetry-protected topological order: emergence of semilocal Gibbs ensembles
• Asymptotic behavior of Toeplitz determinants with a delta function singularity
• Connected correlations in partitioning protocols: a case study and beyond
• No eigenstate of the critical transverse-field Ising chain satisfies the area law