Integrable models, Coulomb interactions, and mean field game theory

Abstract

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 energy levels of heavy nucleii, Riemann zeta zeros, random permutations, and even chicken eyes.

But 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.

This thesis records two threads the author pursued in an attempt to use rigorous mathematics

to understand better how such statistics can emerge.

One thread pursued some technical aspects from the perspective of the field of Integrable

Probability, which can realize such systems as projections of certain representation

theoretic objects, a connection observed in a non-intersecting Poisson

model of the entrepreneurial bus system.

The other thread focuses on the decentralized manner such statistics can

emerge. We accordingly 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.