Dipolar SLEs

TL;DR

This paper introduces Dipolar SLEs, a new stochastic process modeling critical interfaces ending on boundary intervals, linking probabilistic properties with conformal field theory and confirming results through numerical simulations.

Contribution

It develops the theory of Dipolar SLEs, deriving explicit probabilities and their harmonic nature, and connects these with CFT, supported by numerical simulations of the Ising model.

Findings

Derived explicit harmonic probabilities for Dipolar SLEs.

Established a relation between Dipolar SLEs and CFT.

Numerical simulations confirm theoretical predictions.

Abstract

We present basic properties of Dipolar SLEs, a new version of stochastic Loewner evolutions (SLE) in which the critical interfaces end randomly on an interval of the boundary of a planar domain. We present a general argument explaining why correlation functions of models of statistical mechanics are expected to be martingales and we give a relation between dipolar SLEs and CFTs. We compute SLE excursion and/or visiting probabilities, including the probability for a point to be on the left/right of the SLE trace or that to be inside the SLE hull. These functions, which turn out to be harmonic, have a simple CFT interpretation. We also present numerical simulations of the ferromagnetic Ising interface that confirm both the probabilistic approach and the CFT mapping.