Monochromatic path crossing exponents and graph connectivity in two-dimensional percolation.
We consider the fractal dimensions d(k) of the k-connected part of percolation clusters in two dimensions, generalizing the cluster (k=1) and backbone (k=2) dimensions. The codimensions x(k)=2-d(k) describe the asymptotic decay of the probabilities P(r,R) approximately (r/R)(x(k)) that an annulus of radii r<<1 and R>>1 is traversed by k disjoint paths, all living on the percolation clusters. Using a transfer matrix approach, we obtain numerical results for x(k), k<or=6. They are well fitted by the ansatz x(k)=1 / 12k(2)+1 / 48k+(1-k)C, with C=0.0181+/-0.0006.
Jacobsen
Jesper Lykke
JL
Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France. jacobsen@lptms.u-psud.fr
Zinn-Justin
Paul
P