The sharp interface limit of an Ising game

TL;DR

This paper investigates the sharp interface limit of an Ising game, a variant of the Ising model with competing agents, revealing phase transition behavior and deriving a macroscopic interface energy functional.

Contribution

It introduces an Ising game with long-range interactions, analyzes its sharp interface limit, and develops new techniques for mixed local/nonlocal Allen-Cahn type functionals.

Findings

The Ising game exhibits a phase transition with multiple Nash equilibria.

The sharp interface limit leads to a space-time anisotropic perimeter energy.

New methods handle boundary conditions and nonlocal interactions in the limit.

Abstract

The Ising model of statistical physics has served as a keystone example of phase transitions, thermodynamic limits, scaling laws, 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 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.