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

By Dmitry Chelkak, Stanislav Smirnov. 2009

388 citations recorded by the source corpus.

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 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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c200758923ba02ab70cd37f613588a164968b2c9

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Citations in the corpus

Conformal invariance of spin correlations in the planar Ising model (2012, methodology): “The properties of such functions were further investigated in [ChSm12] for a more general class of graphs.”
Discrete complex analysis on isoradial graphs (2008, background): “See [20,3] for the strategy of our proof, [4] for the convergence of certain discrete holomorphic observables and [21] for the square lattice case.”
The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$ (2010, background): “Such observables exist for other models, see [1, 2, 11].”
Convergence of Ising interfaces to Schramm's SLE curves (2013, methodology): “We sketch the proof of Theorem 1 starting with [5, Theorem 1.2], the similar derivation of Theorem 2 from [18, Theorem 2.2] can be found in [8, Section 6.3].”
Conformal loop ensembles: the Markovian characterization and the loop-soup construction (2010, methodology): “The outermost boundaries defined by the union of two independent CLE3s in a domain (recall [7] that CLE3 is the scaling limit of the Ising model loops, and note that it corresponds to c = 1/2) form a CLE4 (which corresponds to “outermost” level lines of the Gaussian Free Field; see [35], [10] and to”