Department of Mathematics

Probability

  •  Dapeng Zhan, MSU
  •  Two-curve Green’s function for 2-SLE
  •  11/15/2018
  •  3:00 PM - 3:50 PM
  •  C405 Wells Hall

A 2-SLE$_\kappa$, $\kappa\in(0,8)$, is a pair of random curves $(\eta_1,\eta_2)$ in a simply connected domain $D$ connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal SLE$_\kappa$ curve in a complement domain. We prove that, for the exponent $\alpha=\frac{(12-\kappa)(\kappa+4)}{8\kappa}$, for any $z_0\in D$, the limit $\lim_{r\to 0^+}r^{-\alpha}\mathbb{P}[\mbox{dist}(\eta_j,z_0)<r,j=1,2]$ converges to a positive number, called the two-curve Green’s function. To prove the convergence, we transform the original problem into the study of a two-dimensional diffusion process, and use orthogonal polynomials to derive its transition density and invariant density.

 

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Michigan State University
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