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We study the number of clusters in 2d critical percolation, NΓ , which intersect a given subset of bonds, Γ. In the simplest case, when Γ is a simple closed curve, NΓ is related to the entanglement entropy of the critical diluted quantum Ising model, in which Γ represents the boundary between the subsystem and the environment. Due to corners in Γ there are universal logarithmic corrections to NΓ , which are calculated in the continuum limit through conformal invariance making use of the Cardy-Peschel formula. The exact formulae are confirmed by large scale Monte Carlo simulations. These results are extended to anisotropic percolation where they confirm a result of discrete holomorphicity.
I. A. Kovács, F. Iglói and J. Cardy, Phys. Rev. B 86, 214203 (2012), arXiv:1210.4671
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