A proof of factorization formula for critical percolation (Q411362)

The present paper is devoted to rigorous proofs of several statements appeared in the papers by \textit{J. J. H. Simmons, P. Kleban} and \textit{R. M. Ziff} [Phys. Rev. E. 76, 041106 (2007); J. Stat. Mech. Theory Exp 2009, P02067 (2009)]. Basically, the proved theorems are related to the factorization of probabilities of several percolation events. Here, percolation means the scaling limit of the critical percolation on the triangular lattice. In the paper, the authors work with a percolation interface having the law of \(\mathrm{SLE}(6)\).