Two-curve Green's function for $2$-SLE: the interior case

A $2$-SLE$_κ$ ($κ\in(0,8)$) is a pair of random curves $(η_1,η_2)$ in a simply connected domain $D$ connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal SLE$_κ$ curve in a complement domain. In this paper we prove that for any $z_0\in D$, the limit $\lim_{r\to 0^+}r^{-α_0} \mathbb{P}[\mbox{dist}(z_0,η_j)<r,j=1,2]$, where $α_0=\frac{(12-κ)(κ+4)}{8κ}$, exists. Such limit is called a two-curve Green's function. We find the convergence rate and the exact formula of the Green's function in terms of a hypergeometric function up to a multiplicative constant. For $κ\in(4,8)$, we also prove the convergence of $\lim_{r\to 0^+}r^{-α_0} \mathbb{P}[\mbox{dist}(z_0,η_1\cap η_2)<r]$, whose limit is a constant times the previous Green's function. To derive these results, we work on two-time-parameter stochastic processes, and use orthogonal polynomials to derive the transition density of a two-dimensional diffusion process that satisfies some system of SDE.

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