Two-Curve Green’s Function for 2-SLE: The Interior Case

A 2- SLE κ ( κ ∈ ( 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 ∈ D , the limit lim r → 0 + r - α 0 P [ dist ( z 0 , η j ) < r , j = 1 , 2 ] , where α 0 = ( 12 - κ ) ( κ + 4 ) 8 κ , 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 ∈ ( 4 , 8 ) , we also prove the convergence of lim r → 0 + r - α 0 P [ dist ( z 0 , η 1 ∩ 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.