### 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 language | English (US) |
---|---|

Pages (from-to) | 1027-1048 |

Number of pages | 22 |

Journal | Journal of Statistical Physics |

Volume | 111 |

Issue number | 5-6 |

DOIs | |

State | Published - Jun 2003 |

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### 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.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 111, no. 5-6, pp. 1027-1048. https://doi.org/10.1023/A:1023006413433

}

TY - JOUR

T1 - On Crossing Event Formulas in Critical Two-Dimensional Percolation

AU - Maier, Robert S

PY - 2003/6

Y1 - 2003/6

N2 - 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.

AB - 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.

KW - Conformal invariance

KW - Critical percolation

KW - Crossing functions

KW - Special functions

KW - Watts's formula

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U2 - 10.1023/A:1023006413433

DO - 10.1023/A:1023006413433

M3 - Article

AN - SCOPUS:0347753689

VL - 111

SP - 1027

EP - 1048

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -