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) Möbius 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.

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)

References in corpus (3)

• Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model
• Critical percolation in the plane
• Approximation of conformal mappings by circle patterns

Cited by in corpus (10)

• Universal finite size corrections and the central charge in non solvable Ising models
• Strong path convergence from Loewner driving function convergence
• A morphological study of cluster dynamics between critical points
• Connection probabilities and RSW-type bounds for the FK Ising model
• Conformal Curves in Potts Model: Numerical Calculation
• Reversibility of Whole-Plane SLE
• The energy density in the planar Ising model
• Critical Ising on the square lattice mixes in polynomial time
• Holomorphic Spinor Observables in the Critical Ising Model
• The critical temperature for the Ising model on planar doubly periodic graphs