{
  "id": "1901.00254",
  "version": "v1",
  "published": "2019-01-02T03:38:16.000Z",
  "updated": "2019-01-02T03:38:16.000Z",
  "title": "Two-curve Green's function for $2$-SLE: the boundary case",
  "authors": [
    "Dapeng Zhan"
  ],
  "comment": "76 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove that for a $2$-SLE$_\\kappa$ pair $(\\eta_1,\\eta_2)$ in a simply connected domain $D$, whose boundary is $C^1$ near $z_0\\in \\partial D$, there is some $\\alpha>0$ such that $\\lim_{r\\to 0^+}r^{-\\alpha} \\mathbb{P}[\\mbox{dist}(z_0,\\eta_j)<r,j=1,2]$ converges to a positive number, called the boundary two-curve Green's function. The exponent $\\alpha$ equals $2(\\frac{12}{\\kappa}-1)$ if $z_0$ is not one of the endpoints of $\\eta_1$ and $\\eta_2$; and otherwise equals $\\frac{12}{\\kappa}-1$. We also prove the existence of the boundary (one-curve) Green's function for a single-boundary-force-point SLE$_\\kappa(\\rho)$ curve, for $\\kappa$ and $\\rho$ in some range. In addition, we find the convergence rate and the exact formula of the above Green's functions up to multiplicative constants. To derive these results, we construct a family of two-dimensional diffusion processes, and use orthogonal polynomials to obtain their transition density.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-02T03:38:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boundary case",
      "boundary two-curve greens function",
      "two-dimensional diffusion processes",
      "single-boundary-force-point sle",
      "exact formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 76,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}