Coupling Independence Implies Zero-Freeness
Abstract
For Δ≥2 and q≥11Δ/6, we prove that the antiferromagnetic q-state Potts partition function on finite simple graphs of maximum degree at most Δ has no Fisher zeros in a graph-uniform complex neighbourhood of [0,1]. For q>11Δ/6, the proof establishes coupling independence throughout [0,1] using a soft version of Vigoda's flip dynamics. Our main tool is a separator-shell transfer theorem for the Potts model on induced-subgraph closed classes of graphs of maximum degree at most Δ, with q≥Δ+1. It yields a graph-uniform zero-free neighbourhood of [0,1] from Hamming coupling independence at $0$ and a uniform coupling-independence bound on each interval [δ,1], δ∈(0,1]. We also obtain zero-free Lee-Yang polydiscs around the uniform field for vertex- and edge-colour fields. For Boolean Holant problems on bounded-degree graphs whose signatures come from a fixed finite family of log-concave symmetric signatures f with f(0)>0, such as b-matchings, we obtain graph-uniform zero-free polytubes around every bounded box of nonnegative activities; their union is an open zero-free neighbourhood of the nonnegative orthant. An appendix summarizes further coupling-independence inputs and the zero-free regions they yield.
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BibTeX
@misc{shao2026coupling,
title = {{Coupling Independence Implies Zero-Freeness}},
author = {Shuai Shao and Ke Shi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12456},
eprint = {2610.12456},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12456},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Coupling Independence Implies Zero-Freeness
AU - Shuai Shao
AU - Ke Shi
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12456
UR - https://arxiv.org/abs/2610.12456
AB - For Δ≥2 and q≥11Δ/6, we prove that the antiferromagnetic q-state Potts partition function on finite simple graphs of maximum degree at most Δ has no Fisher zeros in a graph-uniform complex neighbourhood of [0,1]. For q>11Δ/6, the proof establishes coupling independence throughout [0,1] using a soft version of Vigoda's flip dynamics. Our main tool is a separator-shell transfer theorem for the Potts model on induced-subgraph closed classes of graphs of maximum degree at most Δ, with q≥Δ+1. It yields a graph-uniform zero-free neighbourhood of [0,1] from Hamming coupling independence at $0$ and a uniform coupling-independence bound on each interval [δ,1], δ∈(0,1]. We also obtain zero-free Lee-Yang polydiscs around the uniform field for vertex- and edge-colour fields. For Boolean Holant problems on bounded-degree graphs whose signatures come from a fixed finite family of log-concave symmetric signatures f with f(0)>0, such as b-matchings, we obtain graph-uniform zero-free polytubes around every bounded box of nonnegative activities; their union is an open zero-free neighbourhood of the nonnegative orthant. An appendix summarizes further coupling-independence inputs and the zero-free regions they yield.
ER -