Order and Disorder in Interacting Spin Systems and Random Networks

该研究项目旨在开发概率论中的新工具，以增强对相互作用的粒子系统和随机网络的理解。激励问题涉及基本问题，如马尔可夫链的初始条件（例如，有序或无序，随机或固定）对其收敛到平衡时间的影响，以及各种条件下的结构形成（表面形状，网络中的子图等）与无序。尽管广泛的研究提供了详细的启发和确凿的实验，但许多正在研究的问题仍然是开放的；该项目的目标是通过新的分析方法在这些方面取得进展，这些方法有望在概率论以及其他领域找到进一步的应用。第一个研究项目侧重于自旋系统（如Ising或Potts模型）与自然马尔可夫链（如Metropolis或热浴格劳伯动力学）之间的相互作用。动态和静态模型所经历的相变已经受到了广泛的关注，然而各种基本问题到目前为止还无法在所有三种（高、低和临界）温度下进行严格的分析。本项目研究了几个这样的问题，以及最新进展提出的新问题，包括了解初始构型（随机或确定性，平衡或交替等）对高温混合时间的影响，以及建立3维及以上临界混合的幂律。第二个研究方向侧重于随机表面模型及其演化，对伊辛模型和晶体中的粗化转变产生影响。最后的研究主题是Erdos-Renyi随机图上的随机漫步：该项目旨在研究初始状态对随机漫步混合时间的影响，并在不同的制度下建立典型的结构性质，如其特征对噪声的鲁棒性，以及非典型的结构性质，如当非典型事件（大偏差）为条件时，系统是否组织成不对称结构。

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).