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

相互作用粒子系统的动态演化：混合时间、界面波动和普遍性

• 批准号: 1812095
• 负责人: Eyal Lubetzky
• 金额: $ 33万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-06-01 至 2022-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812095&HistoricalAwards=false
• 关键词: Dynamical; Evolution; Interacting; Particle; Systems

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.

该研究项目旨在研究相互作用粒子系统动力学方面的一系列问题，并在途中开发概率论的新工具，以促进我们对其平衡和非平衡特征的理解。激发问题-其中一些已经在统计和数学物理以及计算机科学中被深入研究-触及基本问题，例如在不同边界条件下的经典模型如2D和3D Ising和Potts模型在相变临界点处的平衡收敛时间，平衡时的团簇界面，（这两个有趣的本身雅阁，并至关重要的分析的动力学），和作用的基础几何的动力学，或缺乏-从而动力学的特点是普遍的一大类图。第一个研究方向重点关注d维环面上的经典Ising、Potts和FK模型，并研究用于对其进行采样/建模其演化的两种最常见的马尔可夫链-- Glauber动力学和Swendsen-Wang动力学。动态和静态模型经历的相变受到了广泛的关注，但迄今为止，在所有三个温度制度（高/低/临界）的严格分析，各种基本问题已经达到。PI建议研究几个这样的问题，包括2D极端四色Potts模型在临界状态下混合的幂律与3D中的缓慢混合，以及建立FK模型非临界状态的截止现象。第二个研究方向旨在了解这些模型在各种边界条件下（公元前）的簇界面特性：PI计划在规定的b.c.下研究3D Ising和2D Potts界面的刚度和波动，对于所有q，对于具有正b.c.的3D Ising模型和具有自由b.c.的临界2D q态Potts模型，最后提出的主题比较了晶格上的相互作用粒子系统与其他环境的行为。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是普适的，PI提议为朗之万动力学建立类似的陈述;对于分支布朗运动，PI将研究存在影响粒子的周期环境的最大值定律;对于随机伊辛模型，PI旨在显示对晶格特有的初始条件的敏感性，与典型的随机规则图上的行为相反。该奖项反映了NSF的法定使命，并被认为值得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。