Percolation Crossing Formulas and Conformal Field Theory

Understand

Using conformal field theory, we derive several new crossing formulas 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 $\Pi_h(r)$, Watts' formula for the horizontal-vertical crossing probability $\Pi_{hv}(r)$, and Cardy's formula for the expected number of clusters crossing horizontally $\mathcal{N}_h(r)$.
• The main step in our approach implies the identification of the derivative of one primary operator with another.