发表机构
Albert Einstein Center for Fundamental Physics（阿尔伯特·爱因斯坦基础物理中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过建立伊辛模型重整化群流与一维动力系统的关系，严格证明了远离临界点时其RG流收敛到零温及无穷温不动点，还给出了$\boldsymbol{\text{Z}}^2$上的明确结果并讨论高维推广。
AI 中文摘要
统计力学现代时期的许多严格结果都是通过几何表示获得的，该领域中被广泛研究的工具是格点自旋模型的随机簇表示。统计力学中另一备受关注但缺乏严格结果的领域是重整化群理论。本文研究将这两个概念找到共同基础的思路，以获得关于重整化群流的严格结果。负簇权重的需求削弱了该方法的成功，但我们仍成功建立了远离临界点的最近邻伊辛模型重整化群流的标度极限，与遵循重整化群变换簇连通性的简单一维动力系统之间的关系。这使得我们能对一大类重整化群变换，严格证明该流分别收敛到零温及无穷温不动点。我们将在$\boldsymbol{\text{Z}}^2$上建立明确结果，随后讨论其向更高维推广的可能性。
英文摘要
Many rigorous results in the modern era of statistical mechanics have been obtained through geometrical representations. A much studied tool in this setting is the random cluster representation of lattice spin models. Another area of statistical mechanics that is of great interest but lacking rigorous results is the theory of the renormalization group. This paper investigates the idea to find a common ground between these two concepts in order to obtain rigorous results on the renormalization group flow. A need for negative cluster-weights weakens the success of this approach. Nevertheless, we managed to establish a relation between the scaling limit of the renormalization group flow of the nearest-neighbour Ising model away from criticality with a simple one-dimensional dynamical system that follows the cluster connectivity of the renormalization group transformation. This allows for a rigorous proof of the convergence of this flow to the zero- and infinite-temperature fixed point respectively for a large family of renormalization group transformations. Explicit results will be established on $\mathbb{Z}^2$ followed by a discussion of the available generalizations to higher dimensions.
CommentsComments/Suggestions very welcome! New Version: Restructured the introduction, fixed some small mistakes and added some explanations. 2nd update: Extended the proof of Lemma 4.4 to make the argument clearer. 3. update: Changed the proof of Corollary 4.7 (now Proposition 4.7) which was based on a false assumption in the previous versions
