The sharp interface limit of an Ising game

Abstract

The Ising model of statistical physics has served as a keystone example of phase transitions and many 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 nonlocal 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 recast as a mixed local/nonlocal 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 new techniques to handle a mixture of local derivative terms and nonlocal interactions.  The boundary conditions imposed by the game theoretic considerations also appear as novel terms and require special treatment.