Dynamical Evolution of Interacting Particle Systems: Mixing Times, Interface Fluctuations and Universality

这项研究项目旨在研究相互作用粒子系统的一系列动力学方面的问题，并在开发概率论的新工具的过程中，以促进我们对它们在平衡中和平衡之外的特征的理解。激励问题-其中几个已经在统计和数学物理以及计算机科学中得到了深入的研究-触及了基本问题，例如在不同边界条件下经典模型(例如2D和3D Ising和Potts模型)在相变临界点的收敛时间、平衡时的团簇界面(这两个界面本身都很有趣，并且对动力学分析至关重要)，以及基础几何对动力学的作用或缺乏--因此动力学特征在广泛的图形类别中是通用的。第一个研究方向集中在d维环面上经典的Ising，Potts和FK模型，并考察了两个最常见的用于采样/模拟其演化的马尔可夫链-Glauber动力学和Swendsen-Wang动力学。动态和静态模型所经历的相变已经引起了人们的极大关注，但到目前为止，在所有三个温度区域(高/低/临界)下，各种基本问题都无法进行严格的分析。PI建议研究几个这样的问题，包括二维极值四色Potts模型与三维慢速混合的幂定律，以及建立FK模型非临界混合的截断现象。第二个研究方向旨在了解这些模型中团簇界面在不同边界条件(B.C.)下的性质：PI计划研究规定B.C.下3D Ising和2D Potts界面的刚性和涨落，以显示具有+B.C.的3D Ising模型的快速混合。以及具有自由B.C.的临界二维Q态Potts模型。最后提出的主题将晶格上相互作用的粒子系统的行为与其他环境进行比较。对于SK自旋玻璃模型，静态测量的各种特征在相互作用定律中是普遍存在的，PI建议为朗之万动力学建立一个类似的声明；对于分支布朗运动，PI将在影响粒子的周期性环境存在的情况下研究极大值定律；对于随机伊辛模型，PI旨在显示对格子特殊的初始条件的敏感性，与典型随机规则图上的行为相反。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This research project aims to study a range of problems on dynamical aspects of systems of interacting particles, and en route to develop new tools in Probability Theory to advance our understanding of their features in and out of equilibrium. The motivating problems - several of which have been intensively studied in statistical and mathematical physics as well as in computer science - touch fundamental questions such as the convergence time to equilibrium at the critical point of phase transition for classical models such as 2D and 3D Ising and Potts models under different boundary conditions, cluster interfaces at equilibrium (both interesting on its own accord, and crucial for the analysis of the dynamics), and the role of the underlying geometry on the dynamics, or the lack thereof - whereby dynamical features are universal on a broad class of graphs. The first research direction focuses on the classical Ising, Potts and FK model on d-dimensional tori, and examines two of the most common Markov chains used to sample them/model their evolution - Glauber dynamics and Swendsen-Wang dynamics. The phase transition that both the dynamical and static models undergo has received much attention, yet various basic problems have so far been out of reach of rigorous analysis in all three temperature regimes (high/low/critical). The PI proposes to study several such problems, including a power-law for mixing at criticality for the extremal 4-color Potts model in 2D vs. slow mixing in 3D, and establishing the cutoff phenomenon for the FK model off-criticality. A second research direction aims to understand properties of cluster interfaces in these models under various boundary conditions (b.c.): The PI plans to study rigidity and fluctuations of 3D Ising and 2D Potts interfaces under prescribed b.c., towards showing fast mixing for the 3D Ising model with plus b.c. and critical 2D q-state Potts model with free b.c. for all q. The final proposed topic compares the behavior of systems of interacting particles on lattices to other environments. For the SK spin glass model, various features of the static measure are known to be universal in the law of the interactions, and the PI proposes to establish an analogous statement for Langevin dynamics; for branching Brownian motion, the PI will investigate the law of the maximum in the presence of a periodic environment affecting the particles; and for the stochastic Ising model, the PI aims to show an aspect of sensitivity to initial conditions that is special to lattices, contrary to the behavior on typical random regular graphs.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.