Critical percolation: the expected number of clusters in a rectangle

TL;DR

This paper introduces new conformally invariant observables for critical site percolation on the triangular lattice, providing explicit limits for the expected number of clusters separating pairs of points, and offers a novel proof approach independent of previous methods.

Contribution

It presents two new observables with conformally invariant scaling limits, advancing understanding of percolation and offering a new proof technique.

Findings

Expected number of clusters converges to a conformal invariant.

New observables are shown to have conformally invariant scaling limits.

Proof is independent of SLE techniques, suggesting alternative approaches.

Abstract

We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and $SLE$ techniques, and in principle should provide a new approach to establishing conformal invariance of percolation.