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

该研究项目旨在研究相互作用粒子系统的动力学方面的一系列问题，并在概率论中开发新的工具，以促进我们对它们在平衡和非平衡状态下的特征的理解。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进行了深入研究-触及基本问题，例如在不同边界条件下二维和三维Ising和Potts模型等经典模型的相变临界点的收敛时间，平衡处的簇界面（两者本身都很有趣，对动力学分析至关重要），以及底层几何在动力学上的作用，或者说是缺乏动力学的作用，因此动力学特征在广泛的图中是通用的。第一个研究方向集中在经典的Ising， Potts和FK模型上的d维环面，并检查了两种最常用的马尔可夫链用于采样/建模它们的演变- Glauber动力学和Swendsen-Wang动力学。动态和静态模型所经历的相变受到了广泛的关注，然而，在所有三种温度状态（高/低/临界）下，各种基本问题迄今为止都无法进行严格的分析。PI建议研究几个这样的问题，包括二维极值四色波茨模型与三维缓慢混合的临界混合的幂律，以及建立非临界FK模型的截止现象。第二个研究方向旨在了解不同边界条件（b.c）下这些模型中的团簇界面的性质：PI计划在规定的b.c下研究3D Ising和2D Potts界面的刚性和波动，以显示具有加b.c的3D Ising模型和具有自由b.c的所有q的临界2D q态Potts模型的快速混合。最后提出的主题是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，已知静态测度的各种特征在相互作用定律中具有普适性，PI提出建立朗之万动力学的类似陈述；对于分支布朗运动，PI将研究存在影响粒子的周期性环境时的最大值定律；对于随机Ising模型，PI旨在显示对初始条件的敏感性，这是格特有的，与典型随机正则图的行为相反。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。

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.