Bridge Decomposition of Restriction Measures

TL;DR

This paper demonstrates that the SLE(8/3) process, which models the scaling limit of the half-plane self-avoiding walk, admits a bridge decomposition derived from its restriction property, generalizing Kesten's discrete results.

Contribution

It establishes a continuum bridge decomposition for SLE(8/3) based on restriction properties, extending Kesten's discrete bridge decomposition to the continuum setting.

Findings

SLE(8/3) admits a bridge decomposition derived from restriction properties.

Restriction hulls with index less than one can be decomposed into a Poisson Point Process of irreducible bridges.

The decomposition is analogous to Ito's excursion decomposition of Brownian motion.

Abstract

Motivated by Kesten's bridge decomposition for two-dimensional self-avoiding walks in the upper half plane, we show that the conjectured scaling limit of the half-plane SAW, the SLE(8/3) process, also has an appropriately defined bridge decomposition. This continuum decomposition turns out to entirely be a consequence of the restriction property of SLE(8/3), and as a result can be generalized to the wider class of restriction measures. Specifically we show that the restriction hulls with index less than one can be decomposed into a Poisson Point Process of irreducible bridges in a way that is similar to Ito's excursion decomposition of a Brownian motion according to its zeros.