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

TL;DR

This paper proves the convergence of fermionic observables in the massive FK-Ising model on isoradial lattices, establishing boundary visit probabilities, connection probabilities, and RSW estimates in the near-critical regime.

Contribution

It introduces a robust method to analyze massive s-holomorphic observables without domain regularity assumptions, advancing understanding of near-critical FK-Ising models.

Findings

Convergence of 2- and 4-point fermionic observables in the massive FK-Ising model.

Establishment of massive RSW-type crossing estimates on isoradial lattices.

Development of techniques exploiting massive s-holomorphicity applicable to broader models.

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 [Smi10], 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. [BeDu12, CIM21, Par19]), these strategies are of direct use in the analysis of massive models in broader setting.