On Crossing Event Formulas in Critical Two-Dimensional Percolation

被引:0
|
作者
Robert S. Maier
机构
[1] University of Arizona,Departments of Mathematics and Physics
来源
Journal of Statistical Physics | 2003年 / 111卷
关键词
Critical percolation; conformal invariance; crossing functions; Watts's formula; special functions;
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摘要
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.
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页码:1027 / 1048
页数:21
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