On Crossing Event Formulas in Critical Two-Dimensional Percolation

2002/10/31 by Robert S. Maier · 2 citations
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.CA #math.CV #math.MP

paper · pdf · doi:10.1023/a:1023006413433

published as J. Statistical Physics 111 (2003) 1027-1048 · final version, accepted by J. Statistical Physics; 22 pages

arxiv created 2002/12/10 · openalex publication_date 2003/04/07 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/29

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 under the assumption of conformal invariance, 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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