{
  "id": "2301.00851",
  "version": "v1",
  "published": "2023-01-02T19:35:23.000Z",
  "updated": "2023-01-02T19:35:23.000Z",
  "title": "The sharp interface limit of an Ising Game",
  "authors": [
    "William M Feldman",
    "Inwon C Kim",
    "Aaron Zeff Palmer"
  ],
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-02T19:35:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sharp interface limit",
      "ising game",
      "perimeter type energy functional",
      "anisotropic perimeter type energy",
      "allen-cahn type minimization problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}