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

自旋系统是由核磁矩（自旋）的相互作用控制的材料的铁磁性和其他性质的数学模型。该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的各种问题。具体研究的模型包括三维伊辛模型（统计力学中最基本的模型之一）和（2+1）维固体-固体模型。这些模型表现出的一个特征是熵排斥，其中最初平坦的表面在其朝向平衡建立高度时经历一系列亚稳态。该项目的主要重点是研究这些模型的熵排斥现象及其与Glauber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，该项目将开发概率论的新方法，这些方法将在相互作用的自旋系统的其他研究中找到应用。该项目为研究生提供了研究训练的机会。第一个研究项目的重点是在一个无限长的圆柱体上的低温下的3D伊辛模型，具有混合边界条件-在高度0以上为负，在其他地方为正。这些产生了一个界面，一个随机的表面分离的正/负相，和PI的目的是研究的熵排斥效应的条件下，这个接口是积极的。主要目标是表明，为了获得熵，表面逐渐上升，其水平线最终形成一个单一的平台，a.s.尺度限制由一个伍尔夫形状;水平线表现出立方根波动远离边界;并开始在一个平坦的初始状态，朝着平衡的表面的演变经历了一系列的亚稳定高原，每个等待时间的双指数在其高度。一个相关的研究方向旨在研究一类近似于3D伊辛的晶体模型，并表明武尔夫形状和立方根波动的尺度限制对整个类都是通用的。对于（2+1）D固体上固体，目标是改进我们的理解，并获得顶层线波动的顺序。最后一个要研究的主题涉及晶体模型，例如当边界条件倾斜时，在低温下的固体上固体，这已知会导致（否则平坦的）表面变得离域。该奖项的目标是给出高度波动的定量界限，并表明Glauber动力学不再因亚稳态而呈指数级缓慢。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.