Crossing Probabilities in Critical 2-D Percolation and Modular Forms

Details Crossing Probabilities in Critical 2-D Percolation and Modular Forms Crossing Probabilities in Critical 2-D Percolation and Modular Forms

1999-11-04

arxiv.org/abs/cond-mat/9911070v1

Physica A281:242-251,2000

Physics

Condensed Matter

Statistical Mechanics

12 pages,LaTeX, uses file elsart.cls from Elsevier. Submitted to Physica (proceedings of StatPhys-Taiwan 1999)

Scientific paper

10.1016/S0378-4371(00)00035-2

Crossing probabilities for critical 2-D percolation on large but finite lattices have been derived via boundary conformal field theory. These predictions agree very well with numerical results. However, their derivation is heuristic and there is evidence of additional symmetries in the problem. This contribution gives a preliminary examination some unusual modular behavior of these quantities. In particular, the derivatives of the "horizontal" and "horizontal-vertical" crossing probabilities transform as a vector modular form, one component of which is an ordinary modular form and the other the product of a modular form with the integral of a modular form. We include consideration of the interplay between conformal and modular invariance that arises.

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