A generalisation of Schramm's formula for SLE(2)

TL;DR

This paper extends Schramm's formula to SLE(2) in the upper half-plane minus a compact set, explicitly determining the left-passage probability, thus broadening understanding of SLE(2) behavior.

Contribution

It generalizes Schramm's formula for SLE(2) in complex domains, providing explicit formulas for passage probabilities in these settings.

Findings

Explicit left-passage probability formula for SLE(2) in H minus K

Extension of Schramm's formula to more general domains

Enhanced understanding of SLE(2) in complex geometries

Abstract

The scaling limit of planar loop-erased random walks is described by a stochastic Loewner evolution with parameter kappa=2. In this note SLE(2) in the upper half-plane H minus a simply-connected compact subset K of H is studied. As a main result, the left-passage probability with respect to K is explicitly determined.