The fermionic observable and the inverse Kac-Ward operator

The discrete fermionic observable for the FK-Ising model on the square lattice was introduced by Smirnov in [53] (although, as mentioned in [14], similar objects appeared in earlier works). He proved in [54] that the scaling limit of the observable at criticality is given by the solution to a Riemann–Hilbert boundary value problem, and therefore is conformally covariant. A generalization of this result to Ising models defined on a large class of isoradial graphs was obtained by Chelkak and Smirnov in [14], yielding also universality of the scaling limit. Since then, several different types of observables have been proposed for both the random cluster and classical spin Ising model. They were used to prove conformal invariance of important quantities in these models. The scaling limit of the energy density of the critical spin Ising model on the square lattice was computed by Hongler and Smirnov [34]. Existence and conformal invariance of the scaling limits of the magnetization and multi-point spin correlations were established by Chelkak, Hongler and Izyurov [13]. The fermionic observables are also among the tools used by Chelkak et al. to prove convergence of the critical Ising interfaces to SLE curves [12]. Moreover, the observables also proved useful in the off-critical regime and were employed by Beffara and Duminil-Copin [6] to give a new proof of criticality of the self-dual point and to calculate the correlation length in the Ising model on the square lattice. In a more recent work of Hongler, Kytölä and Zahabi [33], the fermionic observables were identified as correlation functions of fermion operators in the transfer matrix formalism for the same model. One also has to mention the relation between the fermionic observable and the inverse Kasteleyn operator which was pointed out by Dubédat [22]. In this chapter, we establish a direct connection between the fermionic observable for the spin Ising model and the inverse Kac–Ward operator. In Section 5.1, we describe properties of the complex weights induced by the Kac–Ward operator on the non-backtracking walks in the graph. We then use loop expansions of the even subgraph generating function from Chapter 2 to express the inverse Kac–Ward operator on a finite graph in terms of a weighted sum over a certain family of subgraphs. We call the resulting formula the fermionic generating function since it bears a strong resemblance to the definitions of the spin fermionic observables from [14, 33, 34]. In Section 5.2,