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

TL;DR

This paper proves that the exploration path in 2D critical bond percolation on the square lattice converges to SLE6, confirming a key conjecture linking percolation models and conformal invariance.

Contribution

It establishes the convergence of the critical bond percolation exploration process on the square lattice to SLE6, extending results from site percolation on the hexagonal lattice.

Findings

Convergence of the exploration path to SLE6 in bond percolation on the square lattice.

Construction of path pairs using transformations and conditioning.

Extension of conformal invariance results from hexagonal to 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.