{
  "id": "2304.14375",
  "version": "v1",
  "published": "2023-04-27T17:46:09.000Z",
  "updated": "2023-04-27T17:46:09.000Z",
  "title": "High moments of the SHE in the clustering regimes",
  "authors": [
    "Li-Cheng Tsai"
  ],
  "comment": "35 pages, 5 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We analyze the high moments of the Stochastic Heat Equation (SHE) via a transformation to the attractive Brownian Particles (BPs), which are Brownian motions interacting via pairwise attractive drift. In those scaling regimes where the particles tend to cluster, we prove a Large Deviation Principle (LDP) for the empirical measure of the attractive BPs. Under what we call the 1-to-$n$ initial-terminal condition, we characterize the unique minimizer of the rate function and relate the minimizer to the spacetime limit shapes of the Kardar--Parisi--Zhang (KPZ) equation in the upper tails. The results of this paper are used in the companion paper Lin and Tsai (2023) to prove an $n$-point, upper-tail LDP for the KPZ equation and to characterize the corresponding spacetime limit shape.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-27T17:46:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "high moments",
      "clustering regimes",
      "stochastic heat equation",
      "companion paper lin",
      "large deviation principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}