Is critical 2D percolation universal?

TL;DR

This paper investigates the universality of conformal invariance in 2D critical percolation, exploring extensions of Smirnov's proof to different lattices and embeddings, aiming to generalize Cardy's formula.

Contribution

It proposes new approaches to extend conformal invariance proofs in 2D critical percolation beyond the triangular lattice, focusing on lattice embeddings.

Findings

Ideas presented are promising for future proofs

Partial progress in extending Smirnov's methods

Highlights challenges in lattice embedding choices

Abstract

The aim of this paper 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 the square lattice); the main question we address is that of the choice of the lattice embedding into the plane which gives rise to conformal invariance in the scaling limit. Even though we were not able to produce a complete proof, we believe that the ideas presented here go in the right direction.