Path Crossing Exponents and the External Perimeter in 2D Percolation

Abstract

2D Percolation path exponents $x_{\ell }^{\mathcal{P}}$ describe probabilities for traversals of annuli by $\ell$ non-overlapping paths, each on either occupied or vacant clusters, with at least one of each type. We relate the probabilities rigorously to amplitudes of $O(N=1)$ models whose exponents, believed to be exact, yield $x_{\ell }^{\mathcal{P}}=({\ell }^2-1)/12$. This extends to half-integers the Saleur--Duplantier exponents for $k=\ell /2$ clusters, yields the exact fractal dimension of the external cluster perimeter, $D_{EP}=2-x_3^{\mathcal{P}}=4/3$, and also explains the absence of narrow gate fjords, as originally found by Grossman and Aharony.

4 pages, 2 figures (EPSF). Revised presentation

References in corpus (1)

• Exact Multifractal Exponents for Two-Dimensional Percolation

Cited by in corpus (58)

• 2D growth processes: SLE and Loewner chains
• Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions
• Conformally Invariant Fractals and Potential Theory
• Random planar curves and Schramm-Loewner evolutions
• Two-Dimensional Critical Percolation: The Full Scaling Limit
• Scaling Limits for Minimal and Random Spanning Trees in Two Dimensions
• The Loewner equation: maps and shapes
• Conformal Random Geometry
• Geometrical vs. Fortuin-Kasteleyn Clusters in the Two-Dimensional $q$-State Potts Model
• Winding angle variance of Fortuin-Kasteleyn contours
• Percolation with long-range correlated disorder
• On monochromatic arm exponents for 2D critical percolation
• Percolation on uniform infinite planar maps
• Bond percolation on isoradial graphs: criticality and universality
• Fractal geometry of critical Potts clusters
• Harmonic Measure and Winding of Conformally Invariant Curves
• Some geometric critical exponents for percolation and the random-cluster model
• Universality for bond percolation in two dimensions
• Geometric structure of percolation clusters
• Zipping and collapse of diblock copolymers
• Quantum Hall transitions: An exact theory based on conformal restriction
• SLE local martingales in logarithmic representations
• Adsorption-like Collapse of Diblock Copolymers
• Harmonic Measure for Percolation and Ising Clusters Including Rare Events
• Loop-Cluster Coupling and Algorithm for Classical Statistical Models
• Critical exponents of planar gradient percolation
• Asynchronously parallelised percolation on distributed machines
• The lowest crossing in 2D critical percolation
• Liouville Quantum Gravity and KPZ
• $N$-cluster correlations in four- and five-dimensional percolation
• Pseudoknots in a Homopolymer
• Nesting statistics in the O(n) loop model on random planar maps
• Beyond Blobs in Percolation Cluster Structure: The Distribution of 3-Blocks at the Percolation Threshold
• Fractal dimensions of the Q-state Potts model for the complete and external hulls
• Conformal invariance of loop ensembles under Kardar-Parisi-Zhang dynamics
• Irregularity of polymer domain boundaries in two-dimensional polymer solution
• Exponents for Hamiltonian paths on random bicubic maps and KPZ
• Statistical Mechanics of Confined Polymer Networks
• Nested Closed Paths in Two-Dimensional Percolation
• Entropy production of entirely diffusional Laplacian transfer and the possible role of fragmentation of the boundaries
• Brownian Loops and Conformal Fields
• Directed Spiral Percolation Hull on the Square and Triangular Lattices
• Universal crossing probability in anisotropic systems
• Limit theorems for critical first-passage percolation on the triangular lattice
• The Brownian loop soup stress-energy tensor
• Universality of closed nested paths in two-dimensional percolation
• Basic techniques in two-dimensional critical Ising percolation with investigation of scaling relations
• Numerical studies of planar closed random walks
• Conformal invariance of the loop-erased percolation explorer
• Fractal depth-first search paths in statistical physics models
• One-arm exponent for critical 2D percolation
• Percolation and $O(1)$ loop model
• Conformal invariance and universal critical exponents in the two-dimensional percolation model
• Continuum Nonsimple Loops and 2D Critical Percolation
• A singularity at the criticality for the free energy in percolation
• Hamiltonian cycles on bicolored random planar maps
• The variance of the incipient infinite cluster in two-dimensional percolation
• Critical exponents for two-dimensional percolation