On Crossing Event Formulas in Critical Two-Dimensional Percolation
نویسنده
چکیده
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.
منابع مشابه
2 7 Ju l 2 00 5 The distribution of the minimum height among pivotal sites in critical two - dimensional percolation ∗
Let Ln denote the lowest crossing of the 2n × 2n square box B(n) centered at the origin for critical site percolation on Z 2 or critical site percolation on the triangular lattice imbedded in Z 2 , and denote by Qn the set of pivotal sites along this crossing. On the event that a pivotal site exists, denote the minimum height that a pivotal site attains above the bottom of B(n) by Mn := min{m ≥...
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