The Green function for the radial Schramm–Loewner evolution

Abstract

Abstract. We prove the existence of the Green's function for radial SLEκ for κ < 8. Unlike the chordal case where an explicit formula for the Green's function is known for all values of κ < 8, we give an explicit formula only for κ = 4. For other values of κ, we give a formula in terms of an expectation with respect to SLE conditioned to go through a point.

Schramm’s Formula and the Green’s Function for Multiple SLE

“…We complete this step by establishing the existence of the SLE κ (ρ) Green's function when the force point lies on the boundary and ρ belongs to a certain interval. The proof gives a representation formula in terms of an expectation for two-sided radial SLE stopped at its target point; the formula is similar to that obtained in the main result of [2]. In Step 1, we make a prediction for the observable by choosing an appropriate linear combination of the screening integrals such that the leading order term in the asymptotics cancels (thereby matching the asymptotics we expect).…”

“…and c(κ) is a complex constant. By requiring that G satisfy the correct boundary conditions, we arrive at the prediction (2.9) for the Green's function for SLE κ (2). The trickiest step is the determination of the appropriate screening contour γ.…”

Loop-erased random walk

“…where W t is a standard Brownian motion. The Schwarz lemma implies that the conformal radius at time σ r at least r, and hence σ r corresponds to time at most − 1 2 log r + O (1). Hence, this becomes an estimate about the radial Bessel SDE which is standard.…”

“…where G rad D (z, a, 0) denotes the Green's function for radial SLE 2 in D from a to 0. From [1] we have that…”

Multipoint Estimates for Radial and Whole-Plane SLE

“…Existence of the one-point conformal radius Green's function is proven in [2], but an exact form is only found for κ = 4. In this paper, we use one-point estimates found in [2] and follow the strategy in [14] to prove the following multipoint estimate: 8). Let γ be a radial SLE κ in the unit disc D from 1 to 0, let z 1 , .…”