The sharp interface limit of an Ising Game

William M Feldman, Inwon C Kim, Aaron Zeff Palmer

Published 2023-01-02Version 1

The Ising model of statistical physics has served as a keystone example for phase transitions, thermodynamic limits, scaling laws, renormalization group, and many other phenomena and mathematical methods. We introduce and explore an Ising game, a variant of the Ising model that features competing agents influencing the behavior of the spins. With long-range interactions we consider a mean field limit resulting in a non-local potential game at the mesoscopic scale. This game exhibits a phase transition and multiple constant Nash-equilibria in the supercritical regime. Our analysis focuses on a sharp interface limit for which potential minimizing solutions to the Ising game concentrate on two of the constant Nash-equilibria. We show that the mesoscopic problem can be re-cast as a mixed local / non-local space-time Allen-Cahn type minimization problem. We prove, using a $\Gamma$-convergence argument, that the limiting interface minimizes a space-time anisotropic perimeter type energy functional. This macroscopic scale problem could also be viewed as a problem of optimal control of interface motion. Sharp interface limits of Allen-Cahn type functionals have been well studied, we build on that literature with some new techniques to handle a mixture of local derivative terms and non-local interactions. The boundary conditions imposed by the game theoretic considerations also appear as novel terms and require special treatment.