发表机构
University of Science and Technology of China（中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对最大度不超过Δ的有限简单图，证明当q≥11Δ/6时反铁磁q态Potts配分函数在[0,1]的图一致复邻域内无Fisher零点，还得到布尔Holant问题的零自由区域。
AI 中文摘要
对于Δ≥2且q≥11Δ/6，我们证明了最大度不超过Δ的有限简单图上的反铁磁q态Potts配分函数，在[0,1]的图一致复邻域内不存在Fisher零点。当q>11Δ/6时，该证明利用Vigoda翻转动力学的软版本，在整个[0,1]区间内建立了耦合独立性。我们的主要工具是针对最大度不超过Δ的诱导子图闭类图上Potts模型的分离器-壳转移定理，其中q≥Δ+1；该定理从0处的汉明耦合独立性以及每个区间[δ,1]（δ∈(0,1]）上的一致耦合独立性界，得到[0,1]的图一致零自由邻域。我们还针对顶点和边颜色场，在均匀场周围获得了零自由Lee-Yang多圆盘。对于有界度图上的布尔Holant问题，其签名来自固定有限族的对数凹对称签名f且f(0)>0（如b-匹配），我们在每个有界非负活动箱周围获得了图一致零自由多管，它们的并是正卦限的开零自由邻域。附录总结了进一步的耦合独立性输入及其产生的零自由区域。
英文摘要
For $Δ\ge2$ and $q\ge11Δ/6$, 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 $q\geΔ+1$. 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 $[δ,1]$, $δ\in(0,1]$. 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.
Comments40 pages, 2 figures
