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Abstract: We consider the three new crossing probabilities for percolation recentlyfound via conformal field theory by Simmons, Kleban and Ziff. We prove that allthree of them i may be simply expressed in terms of Cardy-s and Watts-crossing probabilities, ii are weakly holomorphic second-order modularforms of weight 0 and a single particular type on the congruence group$\Gamma2$, and iii under some technical assumptions similar to those usedby Kleban and Zagier, are completely determined by their transformation laws.The only physical input in iii is Cardy-s crossing formula, which suggestsan unknown connection between all crossing-type formulas.



Author: Nikolaos Diamantis, Peter Kleban

Source: https://arxiv.org/







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