Convergence of Fermionic Observables in the Massive Planar FK-Ising Model

Abstract

We prove convergence of the 2- and 4-point fermionic observables of the FK-Ising model on simply connected domains discretised by a planar isoradial lattice in massive (near-critical) scaling limit. The former is alternatively known as a (fermionic) martingale observable (MO) for the massive interface, and in particular encapsulates boundary visit probabilties of the interface. The latter encodes connection probabilities in the 4-point alternating (generalised Dobrushin) boundary condition, whose exact convergence is then further analysed to yield crossing estimates for general boundary conditions. Notably, we obtain a massive version of the so-called Russo-Seymour-Welsh (RSW) type estimates on isoradial lattice. These observables satisfy a massive version of s-holomorphicity Smirnov (Ann. Math. 172: 1435-1467, 2007), and we develop robust techniques to exploit this condition which do not require any regularity assumption of the domain or a particular direction of perturbation. Since many other near-critical observables satisfy the same relation (cf. Beffara (Ann. Probab. 40: 2667-2689, 2012), Chelkak ( arXiv:2104.12858 , 2021), Park (Massive Scaling Limit of the Ising Model: Subcritical Analysis and Isomonodromy, 2019)), these strategies are of direct use in the analysis of massive models in broader setting.