Critical Ising on the Square Lattice Mixes in Polynomial Time
Critical Ising on the Square Lattice Mixes in Polynomial Time
复制标题
多项式时间内方格混合的临界 Ising
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
A. Sly
中科院分区：
文献类型：
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作者：
E. Lubetzky;A. Sly
伊辛模型被广泛认为是统计物理学中研究最多的自旋系统模型。本文的重点是其动态（随机）版本，即 1963 年推出的格劳伯动力学，目前是伊辛测度最流行的采样方法。过去三十年的深入研究已经对 $${\mathbb{Z}^2}$$ 上除了临界点之外的任何地方的动力学谱间隙有了严格的理解。虽然伊辛模型的临界行为长期以来一直是物理学家关注的焦点，但随着 SLE、CLE 和研究共形不变系统的新工具的出现，数学家直到最近才对其临界几何有了理解。静态模型和动态模型之间存在丰富的相互作用。在 Ising 的静态相变处，推测动力学会经历临界减速：在高温下，反能隙为 O(1)，在临界 βc 处，其边长为多项式，在低温下，其为指数函数。一系列开创性的论文在 $${\mathbb{Z}^2}$$ 上验证了这一点，但在 β = βc 时，该行为仍然是一个具有挑战性的开放问题。在这里，我们为临界混合建立了第一个严格的多项式上限，从而确认了 $${\mathbb{Z}^2}$$ 中 Ising 模型的临界减速。也就是说，我们证明，在具有任意（例如固定、自由、周期性）边界条件的有限盒子上，β = βc 处的反间隙在边长上是多项式。该证明利用了对伊辛模型的关键 Fortuin-Kasteleyn 表示的缩放极限的最新理解以及来自马尔可夫链分析的经典工具。
The Ising model is widely regarded as the most studied model of spin-systems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the dynamics on $${\mathbb{Z}^2}$$ everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems.A rich interplay exists between the static and dynamic models. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is O(1), at the critical βc it is polynomial in the side-length and at low temperature it is exponential. A seminal series of papers verified this on $${\mathbb{Z}^2}$$ except at β = βc where the behavior remained a challenging open problem.Here we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in $${\mathbb{Z}^2}$$ . Namely, we show that on a finite box with arbitrary (e.g. fixed, free, periodic) boundary conditions, the inverse-gap at β = βc is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of the critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains.