{
  "id": "2306.05166",
  "version": "v1",
  "published": "2023-06-08T12:57:34.000Z",
  "updated": "2023-06-08T12:57:34.000Z",
  "title": "Large $N$ limit and $1/N$ expansion of invariant observables in $O(N)$ linear $σ$-model via SPDE",
  "authors": [
    "Hao Shen",
    "Rongchan Zhu",
    "Xiangchan Zhu"
  ],
  "comment": "53 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.AP",
    "math.MP"
  ],
  "abstract": "In this paper, we continue the study of large $N$ problems for the Wick renormalized linear sigma model, i.e. $N$-component $\\Phi^4$ model, in two spatial dimensions, using stochastic quantization methods and Dyson--Schwinger equations. We identify the large $N$ limiting law of a collection of Wick renormalized $O(N)$ invariant observables. In particular, under a suitable scaling, the quadratic observables converge in the large $N$ limit to a mean-zero (singular) Gaussian field denoted by $\\mathcal{Q}$ with an explicit covariance; and the observables which are renormalized powers of order $2n$ converge in the large $N$ limit to suitably renormalized $n$-th powers of $\\mathcal{Q}$. The quartic interaction term of the model has no effect on the large $N$ limit of the field, but has nontrivial contributions to the limiting law of the observables, and the renormalization of the $n$-th powers of $\\mathcal{Q}$ in the limit has an interesting finite shift from the standard one. Furthermore, we derive the $1/N$ asymtotic expansion for the $k$-point functions of the quadratic observables by employing graph representations and analyzing the order of each graph from Dyson--Schwinger equations. Finally, turning to the stationary solutions to the stochastic quantization equations, with the Ornstein--Uhlenbeck process being the large $N$ limit, we derive here its next order correction in stationarity, as described by an SPDE with the right-hand side having explicit marginal law which involves the above field $\\mathcal{Q}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-08T12:57:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "invariant observables",
      "th powers",
      "wick renormalized linear sigma model",
      "dyson-schwinger equations",
      "quadratic observables converge"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}