{
  "id": "1806.11275",
  "version": "v1",
  "published": "2018-06-29T05:56:57.000Z",
  "updated": "2018-06-29T05:56:57.000Z",
  "title": "Exact persistence exponent for the $2d$-diffusion equation and related Kac polynomials",
  "authors": [
    "Mihail Poplavskyi",
    "Gregory Schehr"
  ],
  "comment": "6 pages + 14 pages of Supplementary material, 4 figures",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.dis-nn",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We compute the persistence for the $2d$-diffusion equation with random initial condition, i.e., the probability $p_0(t)$ that the diffusion field, at a given point ${\\bf x}$ in the plane, has not changed sign up to time $t$. For large $t$, we show that $p_0(t) \\sim t^{-\\theta(2)}$ with $\\theta(2) = 3/16$. Using the connection between the $2d$-diffusion equation and Kac random polynomials, we show that the probability $q_0(n)$ that Kac polynomials, of (even) degree $n$, have no real root decays, for large $n$, as $q_0(n) \\sim n^{-3/4}$. We obtain this result by using yet another connection with the truncated orthogonal ensemble of random matrices. This allows us to compute various properties of the zero-crossings of the diffusing field, equivalently of the real roots of Kac polynomials. Finally, we unveil a precise connection with a fourth model: the semi-infinite Ising spin chain with Glauber dynamics at zero temperature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-29T05:56:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diffusion equation",
      "exact persistence exponent",
      "related kac polynomials",
      "semi-infinite ising spin chain",
      "connection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}