Pivotal, cluster, and interface measures for critical planar percolation

2013/06/13 by Christophe Garban, Gábor Pete, Oded Schramm · 3 citations
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Mathematical Dynamics and Fractals #Mathematics #Measure (data warehouse) #Scaling limit #Percolation (cognitive psychology) #Limit (mathematics) #Scaling #Percolation critical exponents #Statistical physics #Geometry #Critical exponent #Mathematical analysis #Physics #Computer science

paper · pdf · doi:10.1090/s0894-0347-2013-00772-9

openalex publication_date 2013/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22

Abstract

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 SLE₆ 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.

Citations

• Fast convergence to an invariant measure for non-intersecting 3-dimensional Brownian paths
• Scaling relations for 2D-percolation
• Scaling limits of loop-erased random walks and uniform spanning trees
• The natural parametrization for the Schramm-Loewner evolution
• Contour lines of the two-dimensional discrete Gaussian free field
• Proof of the Van den Berg–Kesten Conjecture
• Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits
• Noise sensitivity of boolean functions and applications to percolation
• Basic properties of the natural parametrization for the Schramm-Loewner evolution

Cited by

• Convergence in natural parametrization of random walk frontier
• Convergence of limit shapes for 2D near-critical first-passage percolation
• Schramm–Loewner evolution [wikipedia]