Strong Szeg\H{o} Limit Theorems for Multi-Bordered, Framed, and Multi-Framed Toeplitz Determinants

TL;DR

This paper extends strong Szeg"H{o} limit theorems to multi-bordered, semi-framed, and multi-framed Toeplitz determinants, providing recursive methods and applications in quantum physics and statistical mechanics.

Contribution

It introduces a general framework for strong Szeg"H{o} limit theorems for complex structured Toeplitz determinants, expanding beyond previous single-bordered cases.

Findings

Derived limit theorems for two-bordered and semi-framed Toeplitz determinants.

Established recursive methods using Dodgson condensation for multi-framed determinants.

Linked determinants to applications in entanglement entropy and lattice models.

Abstract

This work provides the general framework for obtaining strong Szeg\H{o} limit theorems for multi-bordered, semi-framed, framed, and multi-framed Toeplitz determinants, extending the results of Basor et al. (2022) beyond the (single) bordered Toeplitz case. For the two-bordered and also the semi-framed Toeplitz determinants, we compute the strong Szeg\H{o} limit theorems associated with certain classes of symbols, and for the $k$-bordered (${k \geq 3}$), framed, and multi-framed Toeplitz determinants we demonstrate the recursive fashion offered by the Dodgson condensation identities via which strong Szeg\H{o} limit theorems can be obtained. One instance of appearance of semi-framed Toeplitz determinants is in calculations related to the entanglement entropy for disjoint subsystems in the XX spin chain (Brightmore et al. (2020) and Jin-Korepin (2011)). In addition, in the recent work Gharakhloo and Liechty (2024) and in an unpublished work of Professor Nicholas Witte, such determinants have found relevance respectively in the study of ensembles of nonintersecting paths and in the study of off-diagonal correlations of the anisotropic square-lattice Ising model. Besides the intrinsic mathematical interest in these structured determinants, the aforementioned applications have further motivated the study of the present work.