{
  "id": "2109.03163",
  "version": "v1",
  "published": "2021-09-07T15:55:17.000Z",
  "updated": "2021-09-07T15:55:17.000Z",
  "title": "Concentration of the complexity of spherical pure $p$-spin models at arbitrary energies",
  "authors": [
    "Eliran Subag",
    "Ofer Zeitouni"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider critical points of the spherical pure $p$-spin spin glass model with Hamiltonian $H_{N}\\left(\\boldsymbol{\\sigma}\\right)=\\frac{1}{N^{\\left(p-1\\right)/2}}\\sum_{i_{1},...,i_{p}=1}^{N}J_{i_{1},...,i_{p}}\\sigma_{i_{1}}\\cdots\\sigma_{i_{p}}$, where $\\boldsymbol{\\sigma}=\\left(\\sigma_{1},...,\\sigma_{N}\\right)\\in \\mathbb{S}^{N-1}:=\\left\\{ \\boldsymbol{\\sigma}\\in\\mathbb{R}^{N}:\\,\\left\\Vert \\boldsymbol{\\sigma}\\right\\Vert _{2}=\\sqrt{N}\\right\\} $ and $J_{i_{1},...,i_{p}}$ are i.i.d standard normal variables. Using a second moment analysis, we prove that for $p\\geq 32$ and any $E>-E_\\infty$, where $E_\\infty$ is the (normalized) ground state, the ratio of the number of critical points $\\boldsymbol{\\sigma}$ with $H_N(\\boldsymbol{\\sigma})\\leq NE$ and its expectation asymptotically concentrates at $1$. This extends to arbitrary $E$ a similar conclusion of [Sub17a].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-07T15:55:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spherical pure",
      "spin models",
      "arbitrary energies",
      "complexity",
      "spin spin glass model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}