Critical percolation: the expected number of clusters in a rectangle

Clément Hongler, Stanislav Smirnov

Published 2009-09-24Version 1

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.