{
  "id": "2408.13796",
  "version": "v1",
  "published": "2024-08-25T10:00:27.000Z",
  "updated": "2024-08-25T10:00:27.000Z",
  "title": "The game behind oriented percolation",
  "authors": [
    "Avelio Sepúlveda",
    "Bruno Ziliotto"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We characterize the critical parameter of oriented percolation on $\\mathbb{Z}^2$ through the value of a zero-sum game. Specifically, we define a zero-sum game on a percolation configuration of $\\mathbb{Z}^2$, where two players move a token along the non-oriented edges of $\\mathbb{Z}^2$, collecting a cost of 1 for each edge that is open, and 0 otherwise. The total cost is given by the limit superior of the average cost. We demonstrate that the value of this game is deterministic and equals 1 if and only if the percolation parameter exceeds $p_c$, the critical exponent of oriented percolation. Additionally, we establish that the value of the game is continuous at $p_c$. Finally, we show that for $p$ close to 0, the value of the game is equal to 0.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-25T10:00:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "91A15"
    ],
    "keywords": [
      "oriented percolation",
      "zero-sum game",
      "percolation parameter exceeds",
      "percolation configuration",
      "average cost"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}