Percolation and $O(1)$ loop model

TL;DR

This paper provides a comprehensive proof of Cardy's formula for critical percolation on the hexagonal lattice, establishing the universal conformally invariant scaling limit of crossing probabilities with a more conceptual approach.

Contribution

It offers a new, more conceptual proof of Cardy's formula that simplifies previous methods and allows for broader generalizations.

Findings

Confirmed the conformal invariance of the scaling limit

Established the universality of crossing probabilities

Provided a more accessible proof method

Abstract

We present an "ultimate" proof of Cardy's formula for the critical percolation on the hexagonal lattice \cite{Smirnov01criticalpercolation}, showing the existence of the universal and conformally invariant scaling limit of crossing probabilities. The new approach is more conceptual, less technically demanding, and is amenable to generalizations.