Mathematical analysis of strongly correlated processes on discrete dynamic structures

这一建议涵盖了大相互作用系统数学分析的广泛主题，大相互作用系统由几个小组件组成，它们在空间和时间上随机相互作用。这个概念出现在许多领域，概率、统计物理和计算机科学文献中研究的范例包括渗流、自旋系统以及随机和动态网络。严格统计力学的发展及其对现代概率论的影响在上个世纪后半叶成为一个显著的成功故事，不仅丰富了这两个领域，同时也刺激和建立了概率论、复杂分析、动力系统等之间的新联系。双方都产生了许多强大的理论和技术，导致对平衡问题的深刻理解，特别是对于微观水平上的局部相互作用导致弱的宏观独立性的系统。同时，理论计算机科学、组合学和非平衡统计物理的需求提供了一大类模型，其中局部微观相互作用要么在宏观水平上产生强相关性，要么在宏观水平上产生强相关性。或者产生非平衡动态，其行为随着时间的推移发生剧烈变化，打破了平稳性和遍历性。这阻碍了当前基于遍历理论和严格的统计力学技术(例如，能量与熵、有限能量和组合论证)的方法的应用，并使我们面临着巨大的挑战。我们的总体目标是发展数学技术来分析这些重要和困难的模型，在这一领域产生开创性的结果，与其他主题建立新的联系，并开辟未来的研究方向。为了朝着这个大目标取得进展，我们将集中讨论四个特定的模型，它们本身就很有趣，并显示出大类系统共同的重要和具有挑战性的特征和现象。

This proposal embraces the broad theme of mathematical analysis of large interacting systems, which consist of several small components that randomly interact with one another over space and time.This concept arises in many fields, and paradigm examples studied in the probability, statistical physics and computer science literature include percolation, spin systems, and random and dynamic networks. The development of rigorous statistical mechanics and its influence on modern probability theory turned into a remarkable success story in the second half of the last century, not only enriching both fields, but at the same time stimulating and establishing new connections between probability theory, complex analysis, dynamical systems etc.Many powerful theories and techniques were produced on both sides, which led to deep understanding of equilibrium problems, in particular for systems whose local interactions at microscopic level give rise to weak ``macroscopic independence''.In parallel, demands from theoretical computer science, combinatorics and non-equilibrium statistical physics offer a large class of models where local microscopic interactions either produce strong correlations at macroscopic levels, or generate non-equilibrium dynamics, whose behavior changes drastically in time, breaking stationarity and ergodicity. This prevents current methods based on ergodic theory and rigorous statistical mechanics techniques (e.g., energy vs. entropy, finite energy and combinatorial arguments) to be applied, and puts us in front of great challenges. Our overall objective is to develop mathematical techniques to analyze such important and difficult models, producing ground-breaking results in this area, establishing new connections with other topics, and opening up future directions of research.In order to make progress towards this broad goal,we will concentrate on four specific models, which are interesting in their own right, and exhibit important and challenging characteristics and phenomena that are common to a large class of systems.