Critical percolation in the plane

TL;DR

This paper investigates the scaling limits and conformal invariance of critical site percolation on the triangular lattice, establishing key invariance properties and deriving explicit formulas for crossing probabilities.

Contribution

It proves conformal invariance of crossing probabilities and the continuum scaling limit, providing rigorous mathematical foundations for critical percolation in the plane.

Findings

Conformal invariance of crossing probabilities established

Explicit calculation of harmonic conformal invariants

Proof of existence and uniqueness of the scaling limit

Abstract

We study scaling limits and conformal invariance of critical site percolation on triangular lattice. We show that some percolation-related quantities are harmonic conformal invariants, and calculate their values in the scaling limit. As a particular case we obtain conformal invariance of the crossing probabilities and Cardy's formula. Then we prove existence, uniqueness, and conformal invariance of the continuum scaling limit.