Structure and Evolution of Low Temperature Spin Systems: Entropic Repulsion and Metastability

自旋系统是由核磁矩（自旋）的相互作用所控制的铁磁性和其他材料性质的数学模型。本项目旨在研究低温下一些最典型的相互作用自旋系统的基本特征。要研究的具体模型包括三维Ising模型（统计力学中最基本的模型之一）和（2+1）维Solid-On-Solid模型。这些模型表现出的一个特征是熵排斥，即一个最初平坦的表面在趋于平衡的过程中会经历一系列的亚稳定状态。该项目的主要重点是研究这些模型的熵排斥现象及其与格劳伯动力学（模拟系统演化的自然随机过程）的相互作用。为了促进对这些问题的理解，该项目将开发新的概率论方法，这些方法将在其他相互作用自旋系统的研究中得到应用。本项目为研究生提供研究训练机会。第一个研究项目集中在低温下无限圆柱体上的三维Ising模型，混合边界条件-高度0以上为负，其他地方为正。这就产生了一个界面，一个分离正负相的随机表面，PI的目的是研究在这个界面为正的条件下的熵排斥效应。主要目的是表明，为了获得熵，表面逐渐上升，其水平面线最终形成一个单一的平台，其标度极限由Wulff形状给出；水准线在远离边界的地方表现出立方根波动；从一个平坦的初始状态开始，表面向平衡的演变经历了一系列的亚稳定高原，每个高原的高度都有双指数的等待时间。一个相关的研究方向是研究一类近似三维Ising的晶体模型，并表明在Wulff形状和立方根波动方面的缩放极限对整个类是普遍的。对于（2+1）D Solid-On-Solid，目标是完善我们的理解并获得顶层线波动的顺序。最后要研究的主题是晶体模型，如在低温下，当边界条件倾斜时，固体对固体的晶体模型，已知这会导致（否则平坦的）表面变得去局部化。这里的目标是给出高度波动的定量界限，并表明由于亚稳定状态，格劳伯动力学不再是指数慢的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Spin systems are mathematical models for ferromagnetism and other properties of materials that are governed by interactions of nuclear magnetic moments (spins). This project aims to study a variety of problems addressing fundamental features of some of the most canonical interacting spin systems at low temperature. Specific models to be studied include the three dimensional Ising model (one of the most fundamental models in statistical mechanics) and the (2+1) dimensional Solid-On-Solid model. One feature that these models exhibit is the entropic repulsion, where an initially flat surface goes through a series of meta-stable states as it builds its height towards equilibrium. The main focus of this project is to study the entropic repulsion phenomenon for these models and its interplay with Glauber dynamics (the natural stochastic process that models the evolution of the system). To advance understanding of these problems the project will develop new methods in probability theory which would find applications in other studies of interacting spins systems. The project provides research training opportunities for graduate students.The first research project focuses on the 3D Ising model at low temperature on an infinite cylinder, with mixed boundary conditions—minus above height 0 and plus elsewhere. These give rise rise to an interface, a random surface separating the plus/minus phases, and the PI aims to study the entropic repulsion effect conditioned on this interface being positive. The main goal is to show that, in order to gain entropy, the surface gradually rises, and its level lines eventually form a single plateau with an a.s. scaling limit given by a Wulff shape; the level line exhibit cube-root fluctuations away from the boundary; and started at a flat initial state, the evolution of the surface towards equilibrium goes through a sequence of meta-stable plateaus, each with a waiting time doubly-exponential in its height. A related research direction aims to study a class of crystal models that approximate 3D Ising, and show that the scaling limit in terms of a Wulff shape and cube-root fluctuations are universal for that entire class. For the (2+1)D Solid-On-Solid, the goal is to refine our understanding and obtain the order of the fluctuations in the top level line. The final topic to be studied concerns crystal models such as Solid-On-Solid at low temperature when the boundary condition is tilted, which is known to cause the (otherwise flat) surface to become de-localized. Here the goal is to give quantitative bounds on the height fluctuations, and show that Glauber dynamics is no longer exponentially slow due to meta-stable states.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.