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.

这个项目将开发和使用几何和概率工具来解决严格的统计力学理论和图形上的博弈论中的问题。在许多经典的技巧游戏中，玩家交替移动。然而，在随机轮换的游戏中，玩家竞标赢得移动权，并且必须在努力赢得游戏的过程中预算资源。分析这种受赌注控制的随机转向游戏，是一个理解如何从长期来预算宝贵资源以保持战略优势的问题；寻求如何玩这个游戏的解决方案，需要在大手大脚支出的短期领土收益和这种挥霍导致的能力下降的长期成本之间取得平衡。赌注管理的随机转向游戏位于概率和几何的交叉点，是首席调查员(PI)将在此项目中探索的几个方向之一。事实上，概率和几何学的技术也将被用来解决几个重要的物理问题，包括被障碍物捕获如何阻碍线性前进的粒子，或者由于无序随机环境中的生长而形成的随机分形如何对随机无序环境的扰动敏感。通过传播、指导和合作，PI将寻求确保这项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本工具的研究生，以及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形博弈论中的几个问题。我们将研究欧几里得格子中超临界无限开团簇中偏压运动的囚禁机制。这项工作将利用Ornstein-Zernike亚临界晶格模型理论中涉及重采样和外科技术的基本工具，该理论将在此项资助下开发。我们将研究随机介质中Kardar-Parisi-Zhang(KPZ)普适生长模型中尺度宇宙物体的分形结构，以及KPZ尺度结构对噪声的敏感性。为达到后一目的，已用于解决临界渗流动力学问题的谱样本等离散调和分析工具将被重新开发用于最后一道渗流模型。新的公式将被证明，表明熟练的随机转向游戏的玩家如何选择花费决定他们当地获胜概率的预算。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。