Universality in the 2D Ising model and conformal invariance of fermionic observables

Abstract

It is widely believed that the celebrated 2D Ising model at criticality has a universal and conformally invariant scaling limit, which is used in deriving many of its properties. However, no mathematical proof of universality and conformal invariance has ever been given, and even physics arguments support (a priori weaker) M\"obius invariance. We introduce discrete holomorphic fermions for the 2D Ising model at criticality on a large family of planar graphs. We show that on bounded domains with appropriate boundary conditions, those have universal and conformally invariant scaling limits, thus proving the universality and conformal invariance conjectures.Comment: 52 pages, 11 figures. Minor changes + two important ones: a) Section 3.4 (a priori Harnack principle for H) added; b) Section 5 (spin-observable convergence) simplified and rewritten (boundary Harnack principle added, solution in the half-plane simplified