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Universality in the 2D Ising model and conformal invariance of fermionic observables. (English) Zbl 1257.82020
The authors introduce discrete holomorphic fermions for the 2D Ising model at criticality on a large family of planar graphs. It is established that on bounded domains with appropriate boundary conditions, those have universal and conformally invariant scaling limits, which proves the universality and conformal invariance conjectures. These results are first steps to understand a widely believed conjecture that the 2D Ising model at criticality has a universal and conformally invariant scaling limit. The paper deals with the definition of Fortuin-Kasteleyn (random cluster) and spin representations of the critical Ising model on isoradial graphs. The authors introduce basic discrete holomorphic observables (holomorphic fermions) satisfying the martingale property with respect to the growing interface and, essentially, they show show that they satisfy a discrete version of the Cauchy-Riemann equation (Propositions 2.2 and 2.5) using some simple combinatorial bijections between the sets of configurations. It is shown that the observables satisfy the stronger “two-points” equation which is called $$s$$-holomorphicity. Further, some properties of $$s$$-holomorphic functions are studied in Section 3. In Section 4, the authors prove the (uniform) convergence of the basic observable in the FK-Ising model to its continuous counterpart. In Section 5, an analogous convergence result is proved for the holomorphic fermion defined for the spin representation of the critical Ising model (Theorem 5.6). In Section 6, they considere the crossing probability formula for the FK-Ising model on discrete quadrilaterals (Theorem 6.1).

##### MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 60K35 Interacting random processes; statistical mechanics type models; percolation theory 30C35 General theory of conformal mappings 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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