On Crossing Event Formulas in Critical Two-Dimensional Percolation

Details On Crossing Event Formulas in Critical Two-Dimensional Percolation On Crossing Event Formulas in Critical Two-Dimensional Percolation

2002-10-07

arxiv.org/abs/math-ph/0210013v4

J. Statistical Physics 111 (2003) 1027-1048

Physics

Mathematical Physics

final version, accepted by J. Statistical Physics; 22 pages

Scientific paper

10.1023/A:1023006413433

Several formulas for crossing functions arising in the continuum limit of critical two-dimensional percolation models are studied. These include Watts's formula for the horizontal-vertical crossing probability and Cardy's new formula for the expected number of crossing clusters. It is shown that under the assumption of conformal invariance, they simplify when the spatial domain is taken to be the interior of an equilateral triangle. The two crossing functions can be expressed in terms of an equianharmonic elliptic function with a triangular rotational symmetry. This suggests that rigorous proofs of Watts's formula and Cardy's new formula will be easiest to construct if the underlying lattice is triangular. The simplification in a triangular domain of Schramm's `bulk Cardy's formula' is also studied.

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