On Crossing Event Formulas in Critical Two-Dimensional Percolation

Research output: Contribution to journalArticle

6 Citations (Scopus)

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 2003

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Triangular
Equilateral triangle
Conformal Invariance
Rotational symmetry
elliptic functions
Elliptic function
Continuum Limit
simplification
triangles
Simplification
invariance
Simplify
Interior
Horizontal
Vertical
continuums
symmetry
Model

Keywords

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

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

On Crossing Event Formulas in Critical Two-Dimensional Percolation. / Maier, Robert S.

In: Journal of Statistical Physics, Vol. 111, No. 5-6, 06.2003, p. 1027-1048.

Research output: Contribution to journalArticle

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