Explicit finite size corrections for Toeplitz determinants and rectangular domino tilings
Abstract
Finite Toeplitz truncation does not, in general, preserve multiplication. For products of m N× N tridiagonal Toeplitz matrices with symbols z+z−1+ah, we make the two boundary corrections in Widom's product formula for finite sections explicit. The method of images gives two Hankel corrections when m≤ N+2, a sharp uniform degree bound. For N≥ m−1, the ratio of the determinant of the matrix product to the Toeplitz determinant with the product symbol reduces, whenever defined, to a determinant of order 2(m−1), which reversal symmetry splits into two determinants of order m−1. A rectangular Schur polynomial gives an exact formula for the Toeplitz determinant with the product symbol, including when roots coincide. For symbols nonzero on the unit circle, classical identities for rational symbols give an exact m× m determinant formula for the ratio, with an explicit confluent form for repeated roots. With ρh the zero of z+z−1+ah in |z|<1, the limit is ∏i<j(1−ρiρj)2 as N→∞ at fixed m,ah. For rectangular domino tilings, these formulas compare the classical Kasteleyn--Temperley--Fisher product with the banded Toeplitz determinant formed from the product symbol. We obtain the first exponentially small correction at fixed width and relate this comparison to the Toeplitz determinant furnished by the dual Cauchy identity.
Related
- Hankel, Toeplitz, Hermitian--Toeplitz Determinants and the Zalcman Functional for a Class of Biholomorphic Mappings on Complex Banach Spaces
- A proof of Karp's conjecture on the zeros of Toeplitz determinants of rising factorial series
- Toeplitz determinants for starlike mappings of order α in several complex variables
- Symmetry reductions and recurrence degrees for banded Toeplitz determinants and permanents
- Constructive recurrences for determinants and permanents of banded Toeplitz matrices
BibTeX
@misc{tierz2026explicit,
title = {{Explicit finite size corrections for Toeplitz determinants and rectangular domino tilings}},
author = {Miguel Tierz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29844},
eprint = {2610.29844},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29844},
}
RIS
TY - UNPB
TI - Explicit finite size corrections for Toeplitz determinants and rectangular domino tilings
AU - Miguel Tierz
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29844
UR - https://arxiv.org/abs/2610.29844
AB - Finite Toeplitz truncation does not, in general, preserve multiplication. For products of m N× N tridiagonal Toeplitz matrices with symbols z+z^−1+a_h, we make the two boundary corrections in Widom's product formula for finite sections explicit. The method of images gives two Hankel corrections when m≤ N+2, a sharp uniform degree bound. For N≥ m−1, the ratio of the determinant of the matrix product to the Toeplitz determinant with the product symbol reduces, whenever defined, to a determinant of order 2(m−1), which reversal symmetry splits into two determinants of order m−1. A rectangular Schur polynomial gives an exact formula for the Toeplitz determinant with the product symbol, including when roots coincide. For symbols nonzero on the unit circle, classical identities for rational symbols give an exact m× m determinant formula for the ratio, with an explicit confluent form for repeated roots. With ρ_h the zero of z+z^−1+a_h in |z|<1, the limit is ∏_i<j(1−ρ_iρ_j)^2 as N→∞ at fixed m,a_h. For rectangular domino tilings, these formulas compare the classical Kasteleyn--Temperley--Fisher product with the banded Toeplitz determinant formed from the product symbol. We obtain the first exponentially small correction at fixed width and relate this comparison to the Toeplitz determinant furnished by the dual Cauchy identity.
ER -