{
  "id": "1106.4194",
  "version": "v1",
  "published": "2011-06-21T13:24:29.000Z",
  "updated": "2011-06-21T13:24:29.000Z",
  "title": "Rank-driven Markov processes",
  "authors": [
    "Michael Grinfeld",
    "Philip A. Knight",
    "Andrew R. Wade"
  ],
  "comment": "32 pages, 2 colour figures",
  "journal": "Journal of Statistical Physics, Vol. 146 (2012), no. 2, p. 378-407",
  "doi": "10.1007/s10955-011-0368-7",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a class of Markovian systems of $N$ elements taking values in $[0,1]$ that evolve in discrete time $t$ via randomized replacement rules based on the ranks of the elements. These rank-driven processes are inspired by variants of the Bak--Sneppen model of evolution, in which the system represents an evolutionary 'fitness landscape' and which is famous as a simple model displaying self-organized criticality. Our main results are concerned with long-time large-$N$ asymptotics for the general model in which, at each time step, $K$ randomly chosen elements are discarded and replaced by independent $U[0,1]$ variables, where the ranks of the elements to be replaced are chosen, independently at each time step, according to a distribution $\\kappa_N$ on $\\{1,2,...,N\\}^K$. Our main results are that, under appropriate conditions on $\\kappa_N$, the system exhibits threshold behaviour at $s^* \\in [0,1]$, where $s^*$ is a function of $\\kappa_N$, and the marginal distribution of a randomly selected element converges to $U[s^*, 1]$ as $t \\to \\infty$ and $N \\to \\infty$. Of this class of models, results in the literature have previously been given for special cases only, namely the 'mean-field' or 'random neighbour' Bak--Sneppen model. Our proofs avoid the heuristic arguments of some of the previous work and use Foster--Lyapunov ideas. Our results extend existing results and establish their natural, more general context. We derive some more specialized results for the particular case where K=2. One of our technical tools is a result on convergence of stationary distributions for families of uniformly ergodic Markov chains on increasing state-spaces, which may be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-06-21T13:24:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J05",
      "60J10",
      "60K35",
      "82B26",
      "92D15"
    ],
    "keywords": [
      "rank-driven markov processes",
      "time step",
      "main results",
      "bak-sneppen model",
      "evolutionary fitness landscape"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Statistical Physics",
      "year": 2012,
      "month": "Jan",
      "volume": 146,
      "number": 2,
      "pages": 378
    },
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012JSP...146..378G"
    }
  }
}