Percolation crossing formulae and conformal field theory

Abstract

Using conformal field theory, we derive several new crossing formulae at the two-dimensional percolation point. High-precision simulation confirms these results. Integrating them gives a unified derivation of Cardy's formula for the horizontal crossing probability Πh(r), Watts' formula for the horizontal-vertical crossing probability Πhv(r) and Cardy's formula for the expected number of clusters crossing horizontally . The main step in our approach implies the identification of the derivative of one primary operator with another. We present operator identities that support this idea and suggest the presence of additional symmetry in c ≤ 0 conformal field theories. © 2007 IOP Publishing Ltd.