Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model

Abstract

We construct discrete holomorphic fermions in the random cluster Ising model at criticality and show that they have conformally covariant scaling limits (as mesh of the lattice tends to zero). In the sequels those observables are used to construct conformally invariant scaling limits of interfaces and identify those with Schramm’s SLE curves. Though the critical Ising model is often cited as a classical example of conformal invariance, it seems that ours is the first paper where it is actually established.