Order and Disorder in Interacting Spin Systems and Random Networks

相互作用的自旋系统和随机网络中的有序和无序

• 批准号: 1513403
• 负责人: Eyal Lubetzky
• 金额: $ 25.54万
• 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: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1513403&HistoricalAwards=false
• 关键词: Order; Disorder; Interacting; Spin; Systems

This research project aims to develop new tools in probability theory that will enhance understanding of interacting particle systems and random networks. The motivating problems touch fundamental questions such as the effect of initial conditions (e.g., ordered or disordered, random or fixed) in Markov chains on their time of convergence to equilibrium, and structure formation (surface shape, subgraphs in networks, etc.) versus disorder under various conditions. Many of the questions under study remain open, despite extensive studies offering detailed heuristics and corroborating experiments; the goal of this project is to make progress on these via new methods of analysis, which are expected to find further applications in probability theory as well as in other areas. The first research project focuses on the interplay between a spin system, such as the Ising or Potts model, and a natural Markov chain modelling its evolution, for example Metropolis or heat-bath Glauber 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 (high, low, and critical) temperature regimes. This project studies several such problems, as well as newer ones suggested by recent advances, including understanding the effect of the initial configurations (random or deterministic, balanced or alternating, etc.) on the mixing time at high temperature and establishing a power-law for mixing at criticality in dimension 3 and higher. A second research direction focuses on random surface models, as well as their evolution, with ramifications for the Ising model and the roughening transition in crystals. The final research topic addresses random walks on the Erdos-Renyi random graph: the project aims to study the effect of the initial state on the mixing time of random walks, and in different regimes to establish typical structural properties, such as robustness of its features to noise, and atypical ones, such as whether the system organizes to asymmetric structures when conditioned on an atypical event (large deviations).

该研究项目旨在开发概率论中的新工具，以增强对相互作用粒子系统和随机网络的理解。激励问题触及基本问题，如初始条件的影响（例如，有序或无序，随机或固定）在马尔可夫链收敛到平衡点的时间，和结构的形成（表面形状，网络中的子图等）。在不同的条件下与紊乱相比较。许多正在研究的问题仍然是开放的，尽管广泛的研究提供了详细的物理学和证实实验;该项目的目标是通过新的分析方法在这些方面取得进展，预计将在概率论和其他领域找到进一步的应用。第一个研究项目的重点是自旋系统之间的相互作用，如伊辛或波茨模型，和自然马尔可夫链模拟其演变，例如大都会或热浴Glauber动力学。相变的动态和静态模型都受到了很大的关注，但各种基本问题，迄今为止，在所有三个（高，低，临界）温度制度的严格分析。 这个项目研究了几个这样的问题，以及最近的进展提出的新问题，包括理解初始配置的影响（随机或确定性，平衡或交替等）。对高温下混合时间的影响，并建立了3维及更高维临界混合的幂律。 第二个研究方向集中在随机表面模型，以及它们的演变，伊辛模型和粗糙化过渡的影响晶体。最后一个研究主题是Erdos-Renyi随机图上的随机游走：该项目旨在研究初始状态对随机游走混合时间的影响，并在不同的制度中建立典型的结构属性，例如其特征对噪声的鲁棒性，以及非典型的，例如系统在非典型事件（大偏差）的条件下是否组织为非对称结构。