The Dyson and Coulomb Games

Speaker: 

Mark Cerenzia, Dickson Instructor, Department of Mathematics, University of Chicago

Description: 

Random matrix statistics emerge in a broad class of strongly correlated systems, with evidence suggesting they can play a universal role comparable to the one Gaussian and Poisson distributions do classically. Indeed, studies have identified these statistics among heavy nuclei, Riemann zeta zeros, random permutations, and even chicken eyes. However, these statistics have also been observed to emerge in decentralized systems, governing the gaps between entrepreneurial buses, parked cars, perched birds, pedestrians, and other forms of traffic. Accordingly, we construct certain N player dynamic games on the line and in the plane that admit Coulomb gas dynamics as a Nash equilibrium and investigate their basic features, many of which are atypical or even new for the literature on many player games. Most notably, we find that the universal local limit of the equilibrium is sensitive to the chosen model of player information in one dimension but not in two dimensions. We also find that with full information, players can achieve game theoretic symmetry through selfish behavior despite non-exchangeability of states, which allows for strong localized convergence of the N-Nash systems to the mean field master equations against locally optimal player ensembles, i.e., those exhibiting the Nash-optimal local limit. 

Event Topic

Mathematical Finance, Stochastic Analysis, and Machine Learning