Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model

By Stanislav Smirnov. 2007

437 citations recorded by the source corpus.

We construct discrete holomorphic observables in the Ising model at criticality and show that they have conformally covariant scaling limits (as mesh of the lattice tends to zero). In the sequel those observables are used to construct conformally invariant scaling limits of interfaces. Though Ising model is often cited as a classical example of conformal invariance, it seems that ours is the first paper where it is actually established.

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

View Paper: https://www.semanticscholar.org/paper/58923aa6ccfca1bad564b2f5b2e4e8b6c580e839

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Citations in the corpus

Universality in the 2D Ising model and conformal invariance of fermionic observables (2009, methodology): “Proof As on the square grid (see [46]), the proof is based on the bijection between configurations which is produced by their local rearrangement at z.”
Towards conformal invariance of 2D lattice models (2007, background): “Below we will outline our proof [39, 40] that for the Ising parameter q = 2 the interface converges to SLE (16/3), see Figure 4.”
Conformal invariance of spin correlations in the planar Ising model (2012, background): “vided into V1 Cδ and Vi Cδ in a chessboard fashion. By deﬁnition, the restriction of an s-holomorphic function Fto V1,i Cδ is real on V1 Cδ and purely imaginary on Vi Cδ . It is not diﬃcult to check ([Smi10]) that this restriction is in fact discrete holomorphic in the most usual sense, that is, for”
Conformal loop ensembles: the Markovian characterization and the loop-soup construction (2010, methodology): “, with disjoint simple loops) include the Ising model and the discrete Gaussian Free Field level lines ([47, 7, 38, 10]).”
Almost sure multifractal spectrum for the tip of an SLE curve (2009, background): “[21] S. Smirnov (2010) Conformal invariance in random cluster models.”
Gaussian multiplicative chaos and applications: A review (2013, neutral): “1 in [140], we have that ∑ z∈∂Dǫ P ⌊ xǫ (Sτ∂Dǫ = z) ln |ǫ(z − ⌊ y ǫ ⌋)| converges uniformly to ∫ ∂D pD(x, z) ln |z − y|dz.”
Lectures on two-dimensional critical percolation (2007, background): “[43] S. Smirnov (2007), Conformal invariance in random cluster models.”
The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1 (2010, methodology): “We mention that the random-cluster models with parameter q = 2 [27] or q very large [23,24] are much better understood.”
Convergence of Ising interfaces to Schramm's SLE curves (2013, methodology): “In this paper, we provide a self-contained framework to establish this stronger convergence both in the spin and the FK case, by combining the setup from [12] with the crossing estimates from [5, 6] and the results on the observables convergence from [18, 5].”