Towards conformal invariance of 2D lattice models

By Stanislav Smirnov. 2007

285 citations recorded by the source corpus.

Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers, ... This has led to numerous exact (but non-rigorous) predictions of their scaling exponents and dimensions. We will discuss how to prove the conformal invariance conjectures, especially in relation to Schramm-Loewner Evolution.

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

View Paper: https://www.semanticscholar.org/paper/a66f3e8fce972ea1a07d4595ffef9ada2e3bd769

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Conformal loop ensembles: the Markovian characterization and the loop-soup construction (2010, methodology): “cently been proved by Chelkak and Smirnov to converge in distribution to an SLE curve (SLE3) when one lets the mesh of the lattice go to zero (and chooses the critical temperature of the Ising model) [46, 7]. Note that SLE describes the law of one particular interface, not the joint law of all inter”
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.”
Critical percolation exploration path and SLE6: a proof of convergence (2006, background): “We note that Smirnov has recently sketched in [33] a proof different from that”
The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$ (2010, background): “Indeed, this observable should converge, when properly rescaled, to a holomorphic martingale, as explained in [7].”
Lectures on two-dimensional critical percolation (2007, methodology): “[44] S. Smirnov, W. Werner (2001), Critical exponents for two-dimensional percolation, Math.”
Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model (2007, background): “These results were announced and the proofs were sketched in [14], where one can find some of the ideas leading to our approach.”
Duality of chordal SLE (2007, background): “, the convergence of critical percolation on triangular lattice to SLE(6) [16], loop-erased random walk (LERW) to SLE(2) [9,18], uniform spanning tree (UST) Peano curve to SLE(8) [9], Gaussian free field contour line to SLE(4) [13], and some Ising models to SLE(3) and SLE(16/3) [15].”
Conformal invariance in the long-range Ising model (2015, background): “Smirnov for nonlocal observables such as interfaces [44], and recently extended to correlation functions of local operators in bounded domains [45].”
Recent advances in percolation theory and its applications (2015, background): “It has been shown that the geometric spin (GS) cluster interfaces as well as FK cluster interfaces in a 2D Ising model strongly converge to the SLE3 and SLE16/3, respectively, in the scaling limit [85, 277, 278].”
Convergence of Ising interfaces to Schramm's SLE curves (2013, methodology): “Schramm showed that those are the only possible conformally invariant scaling limits of interfaces in 2D critical lattice models, and the convergence to SLE was indeed proved in a number of cases, see [13, 17].”
Universality in the 2D Ising model and conformal invariance of fermionic observables (2009, methodology): “It was introduced in [47], although similar objects appeared earlier in Kadanoff and Ceva [22] (without complex weight and boundary problem discussions) and in Mercat [33] (again without discussion of boundary problem and covariance).”
Conformal invariance of spin correlations in the planar Ising model (2012, methodology): “However, it was noted in [Smi06] that one can naturally define the real part of the integral, using the s-holomorphicity of discrete”