Conformal Invariance in Random Cluster Models. I. Holomorphic Fermions in the Ising Model

It is widely believed that many planar lattice models at the critical temperature are conformally invariant in the scaling limit. In particular, the Ising model is often cited as a classical example of conformal invariance which is used in deriving many of its properties. To the best of our knowledge no mathematical proof of this assertion has ever been given. Moreover, most of the physics arguments concern rectangular domains only (like a plane or a strip) or do not take boundary conditions into account. Thus they give (often unrigorous) justification only of the Möbius invariance of the scaling limit, arguably a much weaker property than full conformal invariance. Of course, success of conformal field theory methods in describing the Ising model provides some evidence for the conformal invariance, but it does not offer an explanation or a proof of the latter. It seems that ours is the first paper, where actual conformal invariance of some observables for the Ising model at criticality (in domains with appropriate boundary conditions) is established. Our methods are different from those employed before, and allow us to obtain sharper versions of some of the known results. Moreover they allow the construction of conformally invariant observables in domains with complicated boundary conditions and on Riemann surfaces. Ultimately we will construct conformally invariant scaling limits of interfaces (random cluster boundaries) and identify them with Schramm’s SLE curves and related loop ensembles. These extensions will be discussed in the sequels [15, 16]. Though one can argue whether the scaling limits of interfaces in the Ising model are of physical relevance, their identification opens possibility for computation of correlation functions and other objects of interest in physics. We consider the Fortuin-Kasteleyn random cluster representation of the Ising model on the square lattice δZ2 at the critical temperature. This representation, briefly reviewed below, studies random clusters, which are clusters of the critical percolation performed on the Ising spin clusters at the critical temperature. The spin correlations can be easily related to connectivity properties in the new model. Every configuration can be described by a collection of interfaces (between random clusters and dual random clusters) which are disjoint loops that fill all the edges of the medial lattice. As a conformally invariant observable we construct a “discrete holomorphic fermion”. In a simply connected domain Ω with two boundary points a and b we introduce Dobrushin boundary conditions, which enforce the existence (besides many loops) of an interface running from a to b, see Figure 1. We show that the expectation that this interface passes through a point z taken with fermionic weight (i.e. a passage in the same direction but with a 2π twist has a relative weight −1,