The Annals of Probability

On the scaling limits of planar percolation

Oded Schramm, Stanislav Smirnov, and Christophe Garban

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Abstract

We prove Tsirelson’s conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also suggest a natural construction for the scaling limit of planar percolation, and more generally of any discrete planar model describing connectivity properties.

Article information

Source
Ann. Probab., Volume 39, Number 5 (2011), 1768-1814.

Dates
First available in Project Euclid: 18 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1318940781

Digital Object Identifier
doi:10.1214/11-AOP659

Mathematical Reviews number (MathSciNet)
MR2884873

Zentralblatt MATH identifier
1231.60116

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 82B43: Percolation [See also 60K35] 60G60: Random fields

Keywords
Percolation noise scaling limit

Citation

Schramm, Oded; Smirnov, Stanislav; Garban, Christophe. On the scaling limits of planar percolation. Ann. Probab. 39 (2011), no. 5, 1768--1814. doi:10.1214/11-AOP659. https://projecteuclid.org/euclid.aop/1318940781


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