New percolation crossing formulas and second-order modular forms

Diamantis, Nikolaos; Kleban, Peter

Abstract

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.

Citation

Diamantis, N., & Kleban, P. (2009). New percolation crossing formulas and second-order modular forms. Communications in Number Theory and Physics, 3(4), 677–696. https://doi.org/10.4310/cntp.2009.v3.n4.a4

• Journal Article Type: Article
• Publication Date: 2009
• Deposit Date: Sep 1, 2023
• Journal: Communications in Number Theory and Physics
• Print ISSN: 1931-4523
• Electronic ISSN: 1931-4531
• Publisher: International Press
• Peer Reviewed: Peer Reviewed
• Volume: 3
• Issue: 4
• Pages: 677–696
• DOI: https://doi.org/10.4310/cntp.2009.v3.n4.a4
• Public URL: https://nottingham-repository.worktribe.com/output/23559345
• Publisher URL: https://www.intlpress.com/site/pub/pages/journals/items/cntp/content/vols/0003/0004/a004/index.php