Critical percolation in finite geometries

1991/11/14 by John Cardy, J L Cardy · 36 citations
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #hep-th

paper · pdf · doi:10.1088/0305-4470/25/4/009

published as J.Phys. A25 (1992) L201-L206 · 10 pages

arxiv created 1991/11/14 · openalex publication_date 1992/02/21 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30

Abstract

The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are shown to be invariant not only under changes of scale, but also under mappings of the region which are conformal in the interior and continuous on the boundary. This is a larger invariance than that expected for generic critical systems. Specific predictions are presented for the crossing probability between opposite sides of a rectangle, and are compared with recent numerical work. The agreement is excellent.

Citations

• Infinite conformal symmetry in two-dimensional quantum field theory
• Some generalized order-disorder transformations

Cited by

• Fractal dimensions of the Q-state Potts model for the complete and external hulls
• From percolation to logarithmic conformal field theory
• Nodal Domain Statistics for Quantum Maps, Percolation, and Stochastic Loewner Evolution
• Freezing into stripe states in two-dimensional ferromagnets and crossing probabilities in critical percolation
• Excess number of percolation clusters on the surface of a sphere
• Tails of the crossing probability
• PP-waves and logarithmic conformal field theories
• String backgrounds and LCFT
• Rigorous confidence intervals for critical probabilities
• A modular invariant bulk theory for the \boldsymbolc=0 triplet model
• Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits
• Logarithmic operators and logarithmic conformal field theories
• Crossing on hyperbolic lattices
• Numerical results for crossing, spanning and wrapping in two-dimensional percolation
• Minimal superstrings and loop gas models
• Scaling limits of loop-erased random walks and uniform spanning trees
• A Solution Space for a System of Null-State Partial Differential Equations: Part 1
• Extended chiral algebras and the emergence of SU(2) quantum numbers in the Coulomb gas
• Factorization of percolation density correlation functions for clusters touching the sides of a rectangle
• LOGARITHMIC CONFORMAL FIELD THEORY WITH BOUNDARY
• Bimodule structure of the mixed tensor product over Uqsℓ(2|1) and quantum walled Brauer algebra
• Euler Integrals for Commuting SLEs
• Excursion decompositions for SLE and Watts' crossing formula
• Radford, Drinfeld and Cardy boundary states in the (1,p) logarithmic conformal field models
• A transfer matrix for the backbone exponent of two-dimensional percolation
• A crossing probability for critical percolation in two dimensions
• Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation
• Toward logarithmic extensions of \widehatsℓ (2)_k conformal field models
• Full Logarithmic Conformal Field theory — an Attempt at a Status Report
• Crossing probabilities in critical 2-D percolation and modular forms
• On three-point connectivity in two-dimensional percolation
• Crossing probability and number of crossing clusters in off-critical percolation
• Boundary S-matrix for the integrable q-Potts model
• Conformal partition functions of critical percolation from D 3 thermodynamic Bethe Ansatz equations
• Effective theory and breakdown of conformal symmetry in a long-range quantum chain
• Open spin chain realization of a topological defect in a one-dimensional Ising model: Boundary and bulk symmetry