{
  "id": "2107.04543",
  "version": "v1",
  "published": "2021-07-09T16:59:02.000Z",
  "updated": "2021-07-09T16:59:02.000Z",
  "title": "Metastability for Glauber dynamics on the complete graph with coupling disorder",
  "authors": [
    "Anton Bovier",
    "Frank den Hollander",
    "Saeda Marello"
  ],
  "comment": "40 pages, 6 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the complete graph on $n$ vertices. To each vertex assign an Ising spin that can take the values $-1$ or $+1$. Each spin $i \\in [n]=\\{1,2,\\dots, n\\}$ interacts with a magnetic field $h \\in [0,\\infty)$, while each pair of spins $i,j \\in [n]$ interact with each other at coupling strength $n^{-1} J(i)J(j)$, where $J=(J(i))_{i \\in [n]}$ are i.i.d. non-negative random variables drawn from a prescribed probability distribution $\\mathcal{P}$. Spins flip according to a Metropolis dynamics at inverse temperature $\\beta \\in (0,\\infty)$. We show that there are critical thresholds $\\beta_c$ and $h_c(\\beta)$ such that, in the limit as $n\\to\\infty$, the system exhibits metastable behaviour if and only if $\\beta \\in (\\beta_c, \\infty)$ and $h \\in [0,h_c(\\beta))$. Our main result are sharp asymptotics, up to a multiplicative error $1+o_n(1)$, of the average crossover time from any metastable state to the set of states with lower free energy. We use standard techniques of the potential-theoretic approach to metastability. The leading order term in the asymptotics does not depend on the realisation of $J$, while the correction terms do. The leading order of the correction term is $\\sqrt{n}$ times a centred Gaussian random variable with a complicated variance depending on $\\beta,h,\\mathcal{P}$ and on the metastable state. The critical thresholds $\\beta_c$ and $h_c(\\beta)$ depend on $\\mathcal{P}$, and so does the number of metastable states. We derive an explicit formula for $\\beta_c$ and identify some properties of $\\beta \\mapsto h_c(\\beta)$. Interestingly, the latter is not necessarily monotone, meaning that the metastable crossover may be re-entrant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-09T16:59:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60K37",
      "82B20",
      "82B44",
      "82C44"
    ],
    "keywords": [
      "complete graph",
      "glauber dynamics",
      "coupling disorder",
      "metastability",
      "metastable state"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}