Dipolar stochastic Loewner evolutions

We present basic properties of dipolar stochastic Loewner evolutions, a new version of stochastic Loewner evolutions (SLEs) 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 conformal field theories (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 for being 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.