The Arc Threshold of the Two-Row Ising Torus: Exact Factorization, an Algebraic Critical Coupling, Uniformity in the Circumference, and the Rate of Convergence
Abstract
The Lee-Yang circle theorem places every zero of a ferromagnetic Ising partition polynomial on the unit circle. The arc condition, that every zero lie in the closed left half-plane, turns the circle theorem into a product formula for the characteristic function of the occupation count and makes it usable for local limit theorems. For the two-row periodic Ising torus we give the exact reflection factorization of the row transfer matrix, reduce the crossing at zeta = i to a real cubic, and identify the critical coupling a_c* = 0.8637850566870... as the unique root in (0,1) of an explicit octic. We prove that every a > a_c* satisfies the arc condition simultaneously for all circumferences, that no smaller coupling does, and that the finite thresholds approach a_c* at the rate C/n^2 with an explicit algebraic constant C. The factorization and the 2x2 threshold are formalized in Lean 4 against Mathlib; the zeta = i reduction, uniformity and rate are analysis-level.
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BibTeX
@misc{mandap2026threshold,
title = {{The Arc Threshold of the Two-Row Ising Torus: Exact Factorization, an Algebraic Critical Coupling, Uniformity in the Circumference, and the Rate of Convergence}},
author = {Marco Mandap},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04165},
eprint = {2611.04165},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04165},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Arc Threshold of the Two-Row Ising Torus: Exact Factorization, an Algebraic Critical Coupling, Uniformity in the Circumference, and the Rate of Convergence
AU - Marco Mandap
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04165
UR - https://arxiv.org/abs/2611.04165
AB - The Lee-Yang circle theorem places every zero of a ferromagnetic Ising partition polynomial on the unit circle. The arc condition, that every zero lie in the closed left half-plane, turns the circle theorem into a product formula for the characteristic function of the occupation count and makes it usable for local limit theorems. For the two-row periodic Ising torus we give the exact reflection factorization of the row transfer matrix, reduce the crossing at zeta = i to a real cubic, and identify the critical coupling a_c* = 0.8637850566870... as the unique root in (0,1) of an explicit octic. We prove that every a > a_c* satisfies the arc condition simultaneously for all circumferences, that no smaller coupling does, and that the finite thresholds approach a_c* at the rate C/n^2 with an explicit algebraic constant C. The factorization and the 2x2 threshold are formalized in Lean 4 against Mathlib; the zeta = i reduction, uniformity and rate are analysis-level.
ER -