{
  "id": "1208.2382",
  "version": "v4",
  "published": "2012-08-11T19:47:58.000Z",
  "updated": "2015-01-09T08:01:37.000Z",
  "title": "No zero-crossings for random polynomials and the heat equation",
  "authors": [
    "Amir Dembo",
    "Sumit Mukherjee"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/13-AOP852 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2015, Vol. 43, No. 1, 85-118",
  "doi": "10.1214/13-AOP852",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider random polynomial $\\sum_{i=0}^na_ix^i$ of independent mean-zero normal coefficients $a_i$, whose variance is a regularly varying function (in $i$) of order $\\alpha$. We derive general criteria for continuity of persistence exponents for centered Gaussian processes, and use these to show that such polynomial has no roots in $[0,1]$ with probability $n^{-b_{\\alpha}+o(1)}$, and no roots in $(1,\\infty)$ with probability $n^{-b_0+o(1)}$, hence for $n$ even, it has no real roots with probability $n^{-2b_{\\alpha}-2b_0+o(1)}$. Here, $b_{\\alpha}=0$ when $\\alpha\\le-1$ and otherwise $b_{\\alpha}\\in(0,\\infty)$ is independent of the detailed regularly varying variance function and corresponds to persistence probabilities for an explicit stationary Gaussian process of smooth sample path. Further, making precise the solution $\\phi_d({\\mathbf{x}},t)$ to the $d$-dimensional heat equation initiated by a Gaussian white noise $\\phi_d({\\mathbf{x}},0)$, we confirm that the probability of $\\phi_d({\\mathbf{x}},t)\\neq0$ for all $t\\in[1,T]$, is $T^{-b_{\\alpha}+o(1)}$, for $\\alpha=d/2-1$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-04-16T22:35:55.000Z",
      "abstract": "Consider random polynomial $\\sum_{i=0}^n a_i x^i$ of independent mean-zero normal coefficients $a_i$, whose variance is a regularly varying function (in $i$) of order $\\alpha$. We derive general criteria for continuity of persistence exponents for centered Gaussian processes, and use these to show that such polynomial has no roots in $[0,1]$ with probability $n^{-b_\\alpha+o(1)}$, and no roots in $(1,\\infty)$ with probability $n^{-b_0+o(1)}$, hence for $n$ even, it has no real roots with probability $n^{-2b_\\alpha - 2b_0+o(1)}$. Here $b_\\alpha =0$ when $\\alpha \\le -1$ and otherwise $b_\\alpha \\in (0,\\infty)$ is independent of the detailed regularly varying variance function and corresponds to persistence probabilities for an explicit stationary Gaussian process of smooth sample path. Further, making precise the solution $\\phi_d(x,t)$ to the $d$-dimensional heat equation initiated by a Gaussian white noise $\\phi_d(x,0)$, we confirm that the probability of $\\phi_d(x,t)\\neq 0$ for all $t\\in [1,T]$, is $T^{-b_{\\alpha} + o(1)}$, for $\\alpha=d/2-1$.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-01-09T08:01:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "35K05",
      "26C10",
      "26A12"
    ],
    "keywords": [
      "random polynomial",
      "regularly varying variance function",
      "probability",
      "zero-crossings",
      "independent mean-zero normal coefficients"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.2382D"
    }
  }
}