On Crossing Event Formulas in Critical Two-Dimensional Percolation

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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.

Original languageEnglish (US)
Pages (from-to)1027-1048
Number of pages22
JournalJournal of Statistical Physics
Volume111
Issue number5-6
DOIs
StatePublished - Jun 1 2003

Keywords

  • Conformal invariance
  • Critical percolation
  • Crossing functions
  • Special functions
  • Watts's formula

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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