Smirnov's Observable for Free Boundary Conditions, Interfaces and Crossing Probabilities

We prove convergence results for variants of Smirnov's fermionic observable in the critical planar Ising model in the presence of free boundary conditions. One application of our analysis is a simple proof of a theorem by Hongler and Kytola on convergence of critical Ising interfaces with plus-minus-free boundary conditions to dipolar SLE(3), and a generalization of this result to an arbitrary number of arcs carrying plus, minus or free boundary conditions. Another application is a computation of scaling limits of crossing probabilities in the critical FK-Ising model with an arbitrary number of alternating wired/free boundary arcs. We also deduce a new crossing formula for the spin Ising model.