The Gaussian free field and SLE(4) on doubly connected domains

TL;DR

This paper explores the relationship between the Gaussian free field and SLE(4) in doubly connected domains, defining new processes and boundary conditions, and extending classical formulas to more complex geometries.

Contribution

It introduces a method to define SLE(4) on doubly connected domains with various boundary conditions, extending Schramm's formula and deriving new passage probabilities.

Findings

Defined SLE(4) processes on doubly connected domains with boundary conditions

Extended Schramm's formula to these domains for Dirichlet and Neumann conditions

Derived new passage probabilities interpolating between boundary conditions

Abstract

The level lines of the Gaussian free field are known to be related to SLE(4). It is shown how this relation allows to define chordal SLE(4) processes on doubly connected domains, describing traces that are anchored on one of the two boundary components. The precise nature of the processes depends on the conformally invariant boundary conditions imposed on the second boundary component. Extensions of Schramm's formula to doubly connected domains are given for the standard Dirichlet and Neumann conditions and a relation to first-exit problems for Brownian bridges is established. For the free field compactified at the self-dual radius, the extended symmetry leads to a class of conformally invariant boundary conditions parametrised by elements of SU(2). It is shown how to extend SLE(4) to this setting. This allows for a derivation of new passage probabilities a la Schramm that interpolate continuously from Dirichlet to Neumann conditions.