Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits

By Stanislav Smirnov. 2001

839 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/f7bbae4e8e107be8f17f54bfd5f7d307e2622483

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Conformally invariant scaling limits: an overview and a collection of problems (2006, background): “Smirnov [86] proved Carleson’s form of Cardy’s formula for critical site percolation on the triangular lattice (that is, the same percolation model we have described above) and showed that crossing probabilities between two arcs on the boundary of a simply connected domain are asymptotically conform”
Percolation on uniform infinite planar maps (2013, methodology): “Without aiming at exhaustivity, let us mention the Loop-Erased Random Walk [22], site percolation on the triangular lattice [29], the Ising model of ferromagnetism [10] and its dual FK-Ising representation [30].”
Commutation relations for Schramm‐Loewner evolutions (2004, background): “These questions are connected with work by Oded Schramm and David Wilson, Greg Lawler, and John Cardy; and also by Bauer and Bernard for a more physical interpretation.”
Contour lines of the two-dimensional discrete Gaussian free field (2006, methodology): “Although many random self-avoiding lattice paths from the statistical physics literature are conjectured to have forms of SLE as scaling limits, rigorous proofs have thus far appeared only for a few cases: site percolation cluster boundaries on the hexagonal lattice (SLE(6), [ Sm ]; see also [CN]), ”
Two-Dimensional Critical Percolation: The Full Scaling Limit (2006, methodology): “The work of Schramm [32] and Smirnov [36] has identified the scaling limit of a certain percolation interface with SLE6, providing, along with the work of Lawler-Schramm-Werner [25, 26] and Smirnov-Werner [40], a confirmation of many results in the physics literature, as well as some new results.”
The dimension of the SLE curves (2002, background): “Smirnov [16] proved that SLE 6 is the scaling limit of critical site percolation interfaces on the triangular grid, and Lawler-Schramm-Werner [11] have proved that SLE 2 and SLE 8 are the respective scaling limits of planar loop-erased random walks and uniform Peano curves.”
The Loewner differential equation and slit mappings (2005, neutral): “Smirnov’s [21] recent work on percolation.”
Exploration trees and conformal loop ensembles (2006, background): “this conjecture has been proved in the case xc = 1/2 and n = 1 (which corresponds to critical Bernoulli site percolation) [2, 15].”
Reversibility of chordal SLE (2008, background): “So far it has been proved that the chordal SLE6 is the scaling limit of the explorer line of the site percolation on the triangular lattice with half open and half closed boundary conditions ([13] and [2]); the chordal SLE8 is the scaling limit of UST Peano curve with half free and half wired bounda”
Scale-free percolation (2011, neutral): “Further, percolation in two dimensions has received tremendous attention in the past years, due to the connection to conformal invariance, see e.g. [28].”
Quantitative noise sensitivity and exceptional times for percolation (2005, background): “Smirnov [30], [29] proved the corresponding statement for critical site percolation on the triangular lattice.”
Basic properties of SLE (2001, methodology): “limit is SLE 2. See Figure 9.1. It was also stated there without proof that SLE 6 is the scaling limit of the boundaries of critical percolation clusters, assuming their conformal invariance. Smirnov [Smi01] has recently proved the conformal invariance conjecture for critical percolation on the tria”
Random planar curves and Schramm-Loewner evolutions (2003, background): “Smirnov’s complete detailed proof of Cardy’s formula is contained in [126, 127].”
SLE curves and natural parametrization (2010, background): “ake a limit as above. There are a number of other models that are known to converge to SLEκin the scaling limit using the capacity parametrization: critical site-percolation on the triangular lattice [16], the level lines of the discrete Gaussian free ﬁeld [14], the interfaces of the random cluster ”
Percolation of random nodal lines (2016, neutral): “PERCOLATION OF RANDOM NODAL LINES 139 • Do Bargmann-Fock random nodal lines converge to SLE(6)? In a celebrated paper [43], Smirnov proved that for the triangular lattice, percolation interfaces converge, as the scale of the lattice tends to zero, to SLE(6).”
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].”