New Percolation Crossing Formulas and Second-order Modular Forms

We consider the three crossing probability densities for percolation recently found via conformal field theory [23]. We prove that all three of them (i) may be simply expressed in terms of Cardy’s [4] and Watts’ [24] crossing probabilities, (ii) are (weakly holomorphic) second-order modular forms of weight 0 (and a single particular type) on the congruence… CONTINUE READING