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

Abstract

We prove that for a 2-SLE κ pair (η 1 , η 2 ) in a simply connected domain D, whose boundary is C 1 near z 0 ∈ ∂D, there is some α > 0 such that lim r→0 + r −α P[dist(z 0 , η j ) < r, j = 1, 2] converges to a positive number, called the boundary two-curve Green's function. The exponent α equals 2( 12 κ − 1) if z 0 is not one of the endpoints of η 1 and η 2 ; and otherwise equals 12 κ − 1. We also prove the existence of the boundary (one-curve) Green's function for a single-boundary-force-point SLE κ (ρ) curve, for κ and ρ 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.converges to a positive number, where the exponent α equals α 0 := (12−κ)(κ+4) 8κ . The limit G(z 0 ) is called the (interior) two-curve Green's function for (η 1 , η 2 ). The paper [22] also derived the convergence rate and the exact formula of G(z 0 ) up to an unknown constant.In this paper we study the limit in the case that z 0 ∈ ∂D, assuming that ∂D is C 1 near z 0 , for some suitable exponent α. Below is our first main theorem.Theorem 1.1. Let κ ∈ (0, 8). Let (η 1 , η 2 ) be a 2-SLE κ in a simply connected domain D. Let z 0 ∈ ∂D. Suppose ∂D is C 1 near z 0 . We have the following results. PreliminaryWe first fix some notation. Let H = {z ∈ C : Im z > 0}. For z 0 ∈ C and S ⊂ C, let rad z 0 (S) = sup{|z − z 0 | : z ∈ S ∪ {z 0 }}. If a function f is absolutely continuous on I, and f ′ = g a.e. on I, then we write f ′ ae = g on I. This means that f (x 2 ) − f (x 1 ) = Let K be a non-empty H-hull. Let K doub = K ∪ {z : z ∈ K}, where K is the closure of K, and z is the complex conjugate of z. By Schwarz reflection principle, there is a compact set S K ⊂ R such that g K extends to a conformal map from C \ K doub onto C \ S K . LetExample. Let x 0 ∈ R, r > 0. Then H := {z ∈ H : |z − x 0 | ≤ r} is an H-hull with g H (z) = z + r 2 z−x 0 , hcap(H) = r 2 , a H = x 0 − r, b H = x 0 + r, H doub = {z ∈ C : |z − x 0 | ≤ r}, c H = x 0 − 2r, d H = x 0 + 2r. The next proposition combines Lemmas 5.2 and 5.3 of [28].

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

“…This paper is the follow-up of [25], in which we proved the existence of the two-curve Green's function for 2-SLE κ at an interior point, and obtained the formula for the Green's function up to a multiplicative constant. In the present paper, we will study the case when the interior point is replaced by a boundary point.…”

“…The two-curve Green's function of a 2-SLE κ is defined to be the rescaled limit of the probability that the two curves in the 2-SLE κ both approach a marked point in D. More specifically, it was proved in [25] that, for any κ ∈ (0, 8), if (η 1 , η 2 ) is a 2-SLE κ in D, and z 0 ∈ D, then the limit G(z 0 ) := lim…”

“…The limit G(z 0 ) is called the (interior) two-curve Green's function for (η 1 , η 2 ). The paper [25] also derived the convergence rate and the exact formula for G(z 0 ) up to an unknown constant.…”

“…The paper uses a two-curve technique introduced in [25], which will be described in Section 1.2. Besides [25] and this paper, the technique was recently used in [24] to study the Green's function for the cut points of chordal SLE κ , κ ∈ (4, 8).…”

“…The paper uses a two-curve technique introduced in [25], which will be described in Section 1.2. Besides [25] and this paper, the technique was recently used in [24] to study the Green's function for the cut points of chordal SLE κ , κ ∈ (4, 8). One future application of the technique is the interior two-curve Green's function for a commuting pair of SLE κ (2, ρ) curves (which may arise as a commuting pair of flow lines in the imaginary geometry theory ( [12]) in the case κ = 4, or a commuting pair of Gaussian free field level lines ( [21]) in the case κ = 4, cf.…”