# Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits

@article{Smirnov2001CriticalPI, title={Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits}, author={Stanislav Smirnov}, journal={Comptes Rendus De L Academie Des Sciences Serie I-mathematique}, year={2001}, volume={333}, pages={239-244} }

## 704 Citations

Conformally invariant scaling limits in planar critical percolation

- Mathematics
- 2009

This is an introductory account of the emergence of conformal
invariance in the scaling limit of planar critical percolation. We give
an exposition of Smirnov's theorem (2001) on the conformal…

Conformal Measure Ensembles for Percolation and the FK–Ising Model

- Mathematics, PhysicsSojourns in Probability Theory and Statistical Physics - II
- 2019

Under some general assumptions, we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in…

The scaling limits of near-critical and dynamical percolation

- Mathematics
- 2013

We prove that near-critical percolation and dynamical percolation on the triangular lattice $\eta \mathbb{T}$ have a scaling limit as the mesh $\eta \to 0$, in the "quad-crossing" space $\mathcal{H}$…

The Full Scaling Limit of Two-Dimensional Critical Percolation

- Mathematics
- 2005

We use SLE(6) 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…

Two-dimensional scaling limits via marked nonsimple loops

- Physics
- 2006

Abstract.We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE6 and hence of the related continuum nonsimple loop process that describes macroscopic…

Percolation crossing formulae and conformal field theory

- Mathematics
- 2007

Using conformal field theory, we derive several new crossing formulae at the two-dimensional percolation point. High-precision simulation confirms these results. Integrating them gives a unified…

Is critical 2D percolation universal

- Mathematics
- 2008

The aim of these notes is to explore possible ways of extending Smirnov’s proof of Cardy’s formula for critical site-percolation on the triangular lattice to other cases (such as bond-percolation on…

On Crossing Event Formulas in Critical Two-Dimensional Percolation

- Mathematics
- 2003

Several formulas for crossing functions arising in the continuum limit of critical two-dimensional percolation models are studied. These include Watts's formula for the horizontal-vertical crossing…

Critical percolation: the expected number of clusters in a rectangle

- Mathematics, Physics
- 2009

We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs…

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