Off-Critical SLE(2) and SLE(4): a Field Theory Approach

TL;DR

This paper investigates off-critical perturbations of SLE(2) and SLE(4) using a field theory approach, revealing martingale properties of partition function ratios and correlation functions under mass perturbations.

Contribution

It introduces a field theory framework to analyze off-critical SLE(2) and SLE(4), connecting partition functions and correlation functions with martingale properties.

Findings

Ratios of massive to massless partition functions are local martingales.

Off-critical drifts are proportional to the logarithmic derivatives of these ratios.

Massive correlation functions are also local martingales for the massive interfaces.

Abstract

Using their relationship with the free boson and the free symplectic fermion, we study the off-critical perturbation of SLE(4) and SLE(2) obtained by adding a mass term to the action. We compute the off-critical statistics of the source in the Loewner equation describing the two dimensional interfaces. In these two cases we show that ratios of massive by massless partition functions, expressible as ratios of regularised determinants of massive and massless Laplacians, are (local) martingales for the massless interfaces. The off-critical drifts in the stochastic source of the Loewner equation are proportional to the logarithmic derivative of these ratios. We also show that massive correlation functions are (local) martingales for the massive interfaces. In the case of massive SLE(4), we use this property to prove a factorisation of the free boson measure.