Two-curve Green’s function for 2-SLE: the boundary case

We prove that for ??(0,8), if (?1,?2) is a 2-SLE?pair in simply connected domain D with an analytic boundary point z0, then as r?0+, P[dist(z0,?j)0, which called the two-curve Green’s function. The exponent ? equals 12 ??1 or 2(12 ??1) depending on whether z0 one of endpoints ?1 ?2. also find convergence rate and exact formula function up multiplicative constant. To derive these results, we construct two-dimensional diffusion processes use orthogonal polynomials obtain their transition density.