CRITICAL EXPONENTS FOR TWO-DIMENSIONAL PERCOLATION

By Stanislav Smirnov, Wendelin Werner. 2001

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We show how to combine Kesten's scaling relations, the determination of critical exponents associated to the stochastic Loewner evolution process by Lawler, Schramm, and Werner, and Smirnov's proof of Cardy's formula, in order to determine the existence and value of critical exponents associated to percolation on the triangular lattice.

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

View Paper: https://www.semanticscholar.org/paper/5c1f132724a09ca7375d900cd95d9f0b5a540ace

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Citations in the corpus

Critical percolation exploration path and SLE6: a proof of convergence (2006, methodology): “We are grateful to Vincent Beffara, Greg Lawler, Oded Schramm, Yuri Suhov and Wendelin Werner for various interesting and useful conversations and to Stas Smirnov for communications about work in progress.”
The dimension of the SLE curves (2002, background): “It is known that for critical percolation interfaces (see [17]) and for loop-erased random”
Lectures on two-dimensional critical percolation (2007, methodology): “Werner) paper on loop-erased random walks [26]). • We will follow some SLE(6) computations from [24, 25]. • For how to deduce the critical exponents for critical percolation from the above, we follow [44, 25]. • For how to deduce the exponents associated to near-critical percolation, we use Kesten’s”
On the scaling of the chemical distance in long-range percolation models (2003, background): “The most well understood questions of percolation theory are those concerning the ap p arance and uniqueness of the giant component (Burton and Keane [10]), uniqueness of the critic al point (Aizenman and Barsky [1], Menshikov [20], Grimmett and Marstrand [16]), decay of conn ectivity functions (Cam”
Conformal invariance of planar loop-erased random walks and uniform spanning trees (2001, background): “This was the case for critical site percolation on the triangular grid [Smi01a, Sch01, LSW02a, SW01 ].”
Quantitative noise sensitivity and exceptional times for percolation (2005, methodology): “For k 2, and r 10k, it was proved in [31] that”
SLE coordinate changes (2005, methodology): “The exponent of (n2 − 1)/12 was shown in [ SW01 ].”
Near-critical percolation in two dimensions (2007, background): “…by the Agence Nationale pour la Recherche under the grant ANR-06-BLAN-0058 †Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France ‡Laboratoire de Mathématiques, Bâtiment 425, Université Paris-Sud 11, 91405 Orsay Cedex, France 1562”
Random planar curves and Schramm-Loewner evolutions (2003, methodology): “the convergence of the discrete exploration process to SLE6 (announced in [126]) should be written up [127] soon. For the derivation of formulas and exponents for critical percolation using SLE6, see [124, 92, 128]. 11 What is missing 11.1 A list of ideas We have listed at the end of each chapter a ”
Percolation Theory (2005, background): “The rigorous derivation of the critical exponents for percolation is due to Smirnov and Werner [19]; the dimension of the limiting curve was obtained by Beffara [2].”
The Random-Cluster Model (2002, background): “It is possible to perform calculations on stochastic Löwner evolutions, and in particular to confirm, [104, 134], the values of many critical exponents associated with percolation (or, at least, site percolation on the triangular lattice).”
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.”
Towards conformal invariance of 2D lattice models (2007, neutral): “It was explored in some detail only for percolation [40], where convergence is known and the required (difficult) estimates were already in place thanks to Harry Kesten [19].”
The Lace Expansion and its Applications (2006, background): “The mean-field value of η is zero, η = 5 24 for d = 2 (this has currently been proven only for site percolation on the triangular lattice [191]), and η is conjectured to be negative for 2 < d < 6 [1].”