Communications in Number Theory and Physics

Volume 3 (2009)

Number 4

New percolation crossing formulas and second-order modular forms

Pages: 677 – 696



Nikolaos Diamantis (School of Mathematical Sciences, University of Nottingham, United Kingdom)

Peter Kleban (LASST and Department of Physics and Astronomy, University of Maine, Orono, Maine, U.S.A.)


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 group Γ(2), and (iii) under some technical assumptions (similar to those used in [19]) are completely determined by their transformation laws.

The only physical input in (iii) is Cardy’s crossing formula, which suggests an unknown connection between all crossing-type formulas.

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