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

Abstract

We construct martingale observables for systems of commuting SLE curves by applying screening techniques within the CFT framework recently developed by Kang and Makarov extended to admit multiple SLEs. We illustrate this approach by rigorously establishing explicit formulas for the Green's function and Schramm's formula in the case of two curves growing towards infinity. In the special case when the curves are "fused" and start at the same point, some of the formulas we prove were predicted in the physics literature.

Asymptotic Analysis of Dotsenko–Fateev Integrals

“…By applying the techniques developed in the first part of the paper, we are able to derive asymptotic estimates for the integrals in these examples. The estimates we establish are used in the derivation of the Green's function and Schramm's formula for multiple SLE presented in [11]. We expect similar asymptotic estimates to be needed also in the context of other SLE observables derived via the screening method.…”

“…constructing observables for such systems via the screening method, see [11] for a more complete discussion. SLE κ curves are constructed by solving Loewner's differential equation…”

“…In [11], the screening method is used to derive explicit formulas for two of the most natural SLE observables: the renormalized probability that the system passes infinitesimally near a given point in H (the Green's function) and the probability that the system passes to the right of a given point in H (Schramm's formula). The derivation in [11] proceeds as follows: First, using a CFT description of the multiple SLE system, the screening method is employed to generate explicit "guesses" for the given observables in terms of Dotsenko-Fateev integrals. The "guesses" are then shown to indeed describe the desired probabilities via a sequence of probabilistic arguments.…”

“…The latter arguments rely heavily on appropriate asymptotic estimates for the relevant Dotsenko-Fateev integrals. In the remainder of this paper, we derive the estimates needed in [11].…”

“…This definition of G(z, ξ 1 , ξ 2 ) can be extended to all α > 1 by continuity, see [11]. It follows from (5.2) and (5.3) that the product y 1− 1 α G(z, ξ 1 , ξ 2 ) only depends on the two angles θ 1 and θ 2 defined by θ j := arg(z − ξ j ), j = 1, 2, see [11, Section 6.2.1].…”