Geometric Probability in Statistical Mechanics and Game Theory

统计力学和博弈论中的几何概率

• 批准号: 2153359
• 负责人: Alan Hammond
• 金额: $ 27万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153359&HistoricalAwards=false
• 关键词: Geometric; Probability; Statistical; Mechanics; Game

This project will develop and employ geometric and probabilistic tools to solve problems in the rigorous theory of statistical mechanics and in game theory on graphs. In many classic games of skill, players move alternately. In random-turn games, however, the players bid to win the right to move and must budget resources in their efforts to win the game. Analyzing such stake-governed random-turn games is a matter of understanding how a precious resource should be budgeted in the long-term in order to maintain strategic advantage; seeking a solution to how to play the game involves capturing the balance needed between the short-term territorial gain of spending big and the long-term cost in diminished capability that arises from such profligacy. Stake-governed random-turn games lie at the intersection of probability and geometry and are one of several directions that the Principal Investigator (PI) will explore in this project. Indeed, techniques from probability and geometry will be also used to address several important physical problems, including how trapping by obstacles impedes a linearly progressing particle, or how random fractals formed as a result of growth in a disordered random environment are sensitive to perturbation of that environment by random disorder. By dissemination, mentorship and collaboration, the PI will seek to ensure that the research enhances the mathematical experience and trajectory of junior researchers including graduate students via joint research to develop fundamental tools, and high-school students via coding projects.The PI will develop robust geometric and probabilistic tools in order to elucidate several problems in the rigorous theory of statistical mechanics and in game theory on graphs. The mechanism of trapping of a biased motion in the supercritical infinite open cluster in the Euclidean lattice will be studied. This work will harness basic tools involving resampling and surgical techniques in the Ornstein-Zernike theory of subcritical lattice models that will be developed under this grant. The fractal structure of scaled universal objects in the Kardar-Parisi-Zhang (KPZ) universality class of models of growth in random media will be studied; as will the sensitivity to noise of scaled KPZ structures. For the latter purpose, tools in discrete harmonic analysis, such as the spectral sample, which have been applied to solve problems in dynamics on critical percolation, will be redeveloped for last passage percolation models. Novel formulas will be proved indicating how skilful players of random-turn games on graphs choose to spend budgets that dictate their local win probabilities.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.

该项目将开发并使用几何和概率工具来解决严格的统计力学理论和图博弈论中的问题。在许多经典的技巧游戏中，玩家都是交替移动的。然而，在随机回合游戏中，玩家竞价赢得移动权，并且必须预算资源以赢得游戏。分析这种由权益控制的随机回合博弈需要了解如何长期预算宝贵的资源以保持战略优势；寻求如何玩这个游戏的解决方案涉及到在大笔支出所带来的短期领土收益与这种挥霍造成的能力削弱的长期成本之间取得所需的平衡。权益控制的随机回合游戏位于概率和几何的交叉点，是首席研究员 (PI) 将在该项目中探索的几个方向之一。事实上，概率和几何技术也将用于解决几个重要的物理问题，包括障碍物的捕获如何阻碍线性前进的粒子，或者由于在无序随机环境中生长而形成的随机分形如何对随机无序对该环境的扰动敏感。通过传播、指导和协作，PI 将寻求确保该研究增强初级研究人员的数学经验和轨迹，包括通过联合研究开发基础工具的研究生和通过编码项目的高中生。PI 将开发强大的几何和概率工具，以阐明严格的统计力学理论和图博弈论中的几个问题。 将研究欧几里德晶格中超临界无限疏散团簇中偏置运动的俘获机制。这项工作将利用亚临界晶格模型 Ornstein-Zernike 理论中涉及重采样和手术技术的基本工具，这些工具将在本次资助下开发。 将研究随机介质生长模型的 Kardar-Parisi-Zhang (KPZ) 普适类中缩放普适对象的分形结构；缩放 KPZ 结构对噪声的敏感性也是如此。对于后一个目的，离散谐波分析工具，例如已应用于解决临界渗流动力学问题的光谱样本，将针对最后通道渗流模型进行重新开发。将证明新颖的公式，表明图表上随机回合游戏的熟练玩家如何选择花费决定其本地获胜概率的预算。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。