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Pivotal, cluster, and interface measures for critical planar percolation

Authors: Christophe Garban, Gábor Pete and Oded Schramm
Journal: J. Amer. Math. Soc. 26 (2013), 939-1024
MSC (2010): Primary 60K35, 81T27, 82B27, 82B43; Secondary 60J67, 60D05, 81T40
Published electronically: June 13, 2013
MathSciNet review: 3073882
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This work is the first in a series of papers devoted to the construction and study of scaling limits of dynamical and near-critical planar percolation and related objects like invasion percolation and the Minimal Spanning Tree. We show here that the counting measure on the set of pivotal points of critical site percolation on the triangular grid, normalized appropriately, has a scaling limit, which is a function of the scaling limit of the percolation configuration. We also show that this limit measure is conformally covariant, with exponent 3/4. Similar results hold for the counting measure on macroscopic open clusters (the area measure) and for the counting measure on interfaces (length measure).

Since the aforementioned processes are very much governed by pivotal sites, the construction and properties of the “local time”-like pivotal measure are key results in this project. Another application is that the existence of the limit length measure on the interface is a key step towards constructing the so-called natural time-parametrization of the $\mathrm {SLE}_6$ curve.

The proofs make extensive use of coupling arguments, based on the separation of interfaces phenomenon. This is a very useful tool in planar statistical physics, on which we included a self-contained Appendix. Simple corollaries of our methods include ratio limit theorems for arm probabilities and the rotational invariance of the two-point function.

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Additional Information

Christophe Garban
Affiliation: Ecole Normale Superieure de Lyon, CNRS and UMPA, 46 Allee d’Italie, 69364 Lyon, Cedex 07 France

Gábor Pete
Affiliation: Institute of Mathematics, Technical University of Budapest, 1 Egry József u, Budapest, 1111 Hungary

Oded Schramm
Affiliation: Microsoft Research, December 10, 1961–September 1, 2008

Received by editor(s): November 24, 2010
Received by editor(s) in revised form: April 26, 2012, and February 2, 2013
Published electronically: June 13, 2013
Additional Notes: The first author was partially supported by ANR grant BLAN06-3-134462
The second author was supported by an NSERC Discovery Grant at the University of Toronto and an EU Marie Curie International Incoming Fellowship at the Technical University of Budapest
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.