Conformal invariance of the exploration path in 2-d critical bond percolation in the square lattice

Abstract

In this paper we present the proof of the convergence of the critical bond percolation exploration process on the square lattice to the trace of SLE${}_6$. This is an important conjecture in mathematical physics and probability. The case of critical site percolation on the hexagonal lattice was established in the seminal work of Smirnov via proving Cardy's formula. Our proof uses a series of transformations and conditioning to construct a pair of paths: the $+\partial$CBP and the $-\partial$CBP. The convergence in the site percolation case on the hexagonal lattice allows us to obtain certain estimates on the scaling limit of the $+\partial$CBP and the $-\partial$CBP. By considering a path which is the concatenation of $+\partial$CBPs and $-\partial$CBPs in an alternating manner, we can prove the convergence in the case of bond percolation on the square lattice.

This is a preliminary version. The first two authors attended the Planar Statistical Models workshop in Sanya from Jan 5 to Jan 8, 2013. We received many critical comments from the participants. We will revise the paper and provide a clean proof

References in corpus (4)

• Critical percolation in the plane
• Strong path convergence from Loewner driving function convergence
• Rate of Convergence for Cardy's Formula
• On the Rate of Convergence for Critical Crossing Probabilities