Two-Dimensional Critical Percolation: The Full Scaling Limit

By Federico Camia, Charles M. Newman. 2006

227 citations recorded by the source corpus.

We use SLE6 paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice – that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.

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

View Paper: https://www.semanticscholar.org/paper/506e3c73eab6b3d80648711f83d201bfb4b17be7

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In Condensed Matter Physics

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Citations in the corpus

Quantitative noise sensitivity and exceptional times for percolation (2005, unknown): “(See also [7].”
Contour lines of the two-dimensional discrete Gaussian free field (2006, background): “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 ]), ”
SLE and the free field: partition functions and couplings (2007, background): “For κ ∈ (4, 8), the data of one chordal SLE path is not sufficient to reconstruct the field; it can be expected that reconstruction is possible from a “fuller” version, such as branching SLEκ ([4, 35]), with similar arguments.”
Near-critical percolation in two dimensions (2007, background): “This particularity, first observed by physicists ([41; 9; 10]), has been proved by Smirnov in [46], and later extended by Camia and Newman in [11].”
Imaginary geometry I: interacting SLEs (2012, background): “SLEκ is a one-parameter family of conformally invariant random curves, introduced by Schramm [27] as a candidate for (and later proved to be) the scaling limit of loop erased random walk [16] and the interfaces in critical percolation [2,33].”
Percolation of random nodal lines (2016, unknown): “It is the main tool used in the control of the geometry of critical clusters [2], and a fundamental ingredient in the obtention of Schramm-Loewner Evolution as a scaling limit [39, 14].”
Lectures on two-dimensional critical percolation (2007, methodology): “equires eﬀort, for instance to show that the discrete domain Markov property translates in the limit to the domain Markov property. This is the strategy followed by Federico Camia and Chuck Newman in [9]. We will follow here the somewhat diﬀerent route outlined by Stas Smirnov in [41]. 1This is due ”
Critical percolation exploration path and SLE6: a proof of convergence (2006, methodology): “We start with two lemmas from [8,9], which are consequences of [2] and of standard bounds on the probability of events corresponding to having a certain number of disjoint monochromatic crossings of an annulus (see Lemma 5 of [17], Appendix A of [21], and also [3]).”
Conformal loop ensembles: the Markovian characterization and the loop-soup construction (2010, background): “These are of course also very interesting objects (see [37], [5], [43] for the description of the percolation scaling limit), but they are not the subject of the present paper.”