Geometric Probability in Statistical Mechanics and Game Theory

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

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.