论文标题
两态抗铁磁性自旋系统的最佳混合
Optimal mixing for two-state anti-ferromagnetic spin systems
论文作者
论文摘要
我们证明了一个最佳的$ω（n^{ - 1}）$用于修改的log-sobolev（MLS）常数Glauber动力学的抗ferromagnetic 2-Spin系统，并在树唯一性方面具有$ n $ pertices。具体而言，该最佳MLS结合对于树独特制度中的以下两旋链系统的类别具有： $ \ bullet $所有严格的抗铁磁性两旋旋转系统（其中两个边缘参数$β，γ<1 $），它们涵盖了硬核模型和抗铁磁模型； $ \ bullet $常规图上的常规抗铁磁两旋转系统。 因此，当满足唯一性条件时，这些反铁磁性两旋链系统的最佳$ o（n \ log n）$混合时间将保留。这些MLS和混合时间边界对于任何有限的最大程度或无界最大程度都保持，并且边界中的恒定因子仅取决于唯一性阈值的间隙。我们通过显示出满足某些光谱独立性和边缘稳定性特性的分布的MLS常数的增强定理来证明这一点。
We prove an optimal $Ω(n^{-1})$ lower bound for modified log-Sobolev (MLS) constant of the Glauber dynamics for anti-ferromagnetic two-spin systems with $n$ vertices in the tree uniqueness regime. Specifically, this optimal MLS bound holds for the following classes of two-spin systems in the tree uniqueness regime: $\bullet$ all strictly anti-ferromagnetic two-spin systems (where both edge parameters $β,γ<1$), which cover the hardcore models and the anti-ferromagnetic Ising models; $\bullet$ general anti-ferromagnetic two-spin systems on regular graphs. Consequently, an optimal $O(n\log n)$ mixing time holds for these anti-ferromagnetic two-spin systems when the uniqueness condition is satisfied. These MLS and mixing time bounds hold for any bounded or unbounded maximum degree, and the constant factors in the bounds depend only on the gap to the uniqueness threshold. We prove this by showing a boosting theorem for MLS constant for distributions satisfying certain spectral independence and marginal stability properties.
