{
  "id": "0908.2088",
  "version": "v1",
  "published": "2009-08-14T15:49:12.000Z",
  "updated": "2009-08-14T15:49:12.000Z",
  "title": "Sharp approximation for density dependent Markov chains",
  "authors": [
    "Kamil Szczegot"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a sequence (indexed by n) of Markov chains Z^n in R^d characterized by transition kernels that approximately (in n) depend only on the rescaled state n^{-1} Z^n. Subject to a smoothness condition, such a family can be closely coupled on short time intervals to a Brownian motion with quadratic drift. This construction is used to determine the first two terms in the asymptotic (in n) expansion of the probability that the rescaled chain exits a convex polytope. The constant term and the first correction of size n^{-1/6} admit sharp characterization by solutions to associated differential equations and an absolute constant. The error is smaller than O(n^{-b}) for any b < 1/4. These results are directly applied to the analysis of randomized algorithms at phase transitions. In particular, the `peeling' algorithm in large random hypergraphs, or equivalently the iterative decoding scheme for low-density parity-check codes over the binary erasure channel is studied to determine the finite size scaling behavior for irregular hypergraph ensembles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-08-14T15:49:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60F17",
      "82B26",
      "68W20",
      "94A29"
    ],
    "keywords": [
      "density dependent markov chains",
      "sharp approximation",
      "short time intervals",
      "admit sharp characterization",
      "large random hypergraphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0908.2088S"
    }
  }
}