Geometric Probability in Statistical Mechanics and Game Theory

该项目将开发和使用几何和概率工具来解决统计力学的严格理论和图上的博弈论中的问题。在许多经典技巧游戏中，玩家交替移动。然而，在随机回合游戏中，玩家为了赢得移动的权利而竞标，并且必须为赢得游戏而预算资源。分析这类由风险控制的随机回合游戏需要理解，为了保持战略优势，应该如何对宝贵资源进行长期预算；寻找如何玩这个游戏的解决方案，需要在大笔支出带来的短期领土收益和这种挥霍所带来的能力下降的长期成本之间取得平衡。风险控制的随机回合游戏处于概率和几何的交叉点，是首席调查员（PI）将在该项目中探索的几个方向之一。事实上，来自概率和几何的技术也将用于解决几个重要的物理问题，包括障碍物如何捕获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将寻求确保研究提高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，以阐明统计力学的严格理论和图上的博弈论中的几个问题。研究了欧几里得晶格中超临界无限开团中偏置运动的俘获机制。这项工作将利用亚临界晶格模型的Ornstein-Zernike理论中涉及重采样和手术技术的基本工具，该理论将在该资助下开发。研究了随机介质中生长模型的kardar - paris - zhang （KPZ）普适类中尺度通用物体的分形结构；同样也会影响到尺度KPZ结构对噪声的敏感性。对于后一种目的，离散谐波分析中的工具，如光谱样本，已经应用于解决临界渗流动力学问题，将被重新开发用于最后通道渗流模型。新的公式将被证明，表明随机回合游戏的熟练玩家如何选择决定他们局部获胜概率的预算。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.