Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 062, 51 pages      arXiv:2309.14695      https://doi.org/10.3842/SIGMA.2024.062
Contribution to the Special Issue on Evolution Equations, Exactly Solvable Models and Random Matrices in honor of Alexander Its' 70th birthday

Strong Szegő Limit Theorems for Multi-Bordered, Framed, and Multi-Framed Toeplitz Determinants

Roozbeh Gharakhloo
Mathematics Department, University of California, Santa Cruz, CA 95064, USA

Received September 27, 2023, in final form June 20, 2024; Published online July 11, 2024

Abstract
This work provides the general framework for obtaining strong Szegő 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 Szegő 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 Szegő 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 In addition, in the recent work Gharakhloo and Liechty (2024) and in an unpublished work of Professor Nicholas Witte, such determinants have found relevance respectively in the study of ensembles of nonintersecting paths and in the study of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.

Key words: strong Szegő theorem; bordered Toeplitz determinants; framed Toeplitz determinants; Riemann-Hilbert problem; asymptotic analysis.

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