{
  "id": "2005.09279",
  "version": "v1",
  "published": "2020-05-19T08:27:00.000Z",
  "updated": "2020-05-19T08:27:00.000Z",
  "title": "Large $N$ Limit of the $O(N)$ Linear Sigma Model via Stochastic Quantization",
  "authors": [
    "Hao Shen",
    "Scott Smith",
    "Rognchan Zhu",
    "Xiangchan Zhu"
  ],
  "comment": "72 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.AP",
    "math.MP"
  ],
  "abstract": "This article studies large $N$ limits of a coupled system of $N$ interacting $\\Phi^4$ equations posed over $\\mathbb{T}^{d}$ for $d=1,2$, known as the $O(N)$ linear sigma model. Uniform in $N$ bounds on the dynamics are established, allowing us to show convergence to a mean-field singular SPDE, also proved to be globally well-posed. Moreover, we show tightness of the invariant measures in the large $N$ limit. For large enough mass, they converge to the (massive) Gaussian free field, the unique invariant measure of the mean-field dynamics, at a rate of order $1/\\sqrt{N}$ with respect to the Wasserstein distance. We also consider fluctuations and obtain tightness results for certain $O(N)$ invariant observables, along with an exact description of the limiting correlations in $d=1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-19T08:27:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear sigma model",
      "stochastic quantization",
      "unique invariant measure",
      "article studies large",
      "gaussian free field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 72,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}