$SLE_6$ and 2-d critical bond percolation on the square lattice

TL;DR

This paper proves that the exploration path of 2D critical bond percolation on the square lattice converges to SLE_6, using new observables and harmonic analysis techniques, confirming conformal invariance in the scaling limit.

Contribution

It introduces two new edge observables derived from Smirnov's parafermionic observable, enabling the proof of convergence to SLE_6 on the square lattice.

Findings

Convergence of the exploration path to SLE_6 established.

New edge observables facilitate analysis of percolation interfaces.

Application of harmonic analysis and conformal mapping techniques.

Abstract

Through the rotational invariance of the 2-d critical bond percolation exploration path on the square lattice we express Smirnov's edge parafermionic observable as a sum of two new edge observables. With the help of these two new edge observables we can apply the discrete harmonic analysis and conformal mapping theory to prove the convergence of the 2-d critical bond percolation exploration path on the square lattice to the trace of $SLE_6$ as the mesh size of the lattice tends to zero.