Robert S. Maier
On crossing event formulas in critical two-dimensional percolation
(58K, LATeX2e)

ABSTRACT.  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 for lattices where conformal invariance holds, 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.