{
  "id": "0705.2648",
  "version": "v2",
  "published": "2007-05-18T09:03:05.000Z",
  "updated": "2007-08-07T08:51:07.000Z",
  "title": "Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation",
  "authors": [
    "Gregory Schehr",
    "Satya N. Majumdar"
  ],
  "comment": "4 pages, 3 figures. Minor changes. Accepted version in Phys. Rev. Lett",
  "journal": "Phys. Rev. Lett. 99, 060603 (2007)",
  "doi": "10.1103/PhysRevLett.99.060603",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.dis-nn",
    "math.PR"
  ],
  "abstract": "We consider a class of real random polynomials, indexed by an integer d, of large degree n and focus on the number of real roots of such random polynomials. The probability that such polynomials have no real root in the interval [0,1] decays as a power law n^{-\\theta(d)} where \\theta(d)>0 is the exponent associated to the decay of the persistence probability for the diffusion equation with random initial conditions in space dimension d. For n even, the probability that such polynomials have no root on the full real axis decays as n^{-2(\\theta(d) + \\theta(2))}. For d=1, this connection allows for a physical realization of real random polynomials. We further show that the probability that such polynomials have exactly k real roots in [0,1] has an unusual scaling form given by n^{-\\tilde \\phi(k/\\log n)} where \\tilde \\phi(x) is a universal large deviation function.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-08-07T08:51:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05.40.-a",
      "02.50.-r",
      "05.70.Ln",
      "82.40.Bj"
    ],
    "keywords": [
      "diffusion equation",
      "zero crossings",
      "real random polynomials",
      "real root",
      "full real axis decays"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Physical Review Letters",
      "year": 2007,
      "month": "Aug",
      "volume": 99,
      "number": 6,
      "pages": "060603"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007PhRvL..99f0603S"
    }
  }
}