Dipolar Sles

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. Two types of stochastic Loewner evolutions (SLE) have been thoroughly studied, the chordal and the radial SLEs [1, 2, 3, 4]. The former describe random curves joining two points on the boundary of a simply connected planar domain while the latter describe curves joining one point on the boundary to a point in the bulk of the domain. They correspond to two inequivalent normalizations of conformal maps between simply connected domains in C.