Monochromatic path crossing exponents and graph connectivity in 2D percolation
Abstract
We consider the fractal dimensions dk of the k-connected part of percolation clusters in two dimensions, generalizing the cluster (k=1) and backbone (k=2) dimensions. The codimensions Xk = 2-dk describe the asymptotic decay of the probabilities P(r,R) ~ (r/R)Xk 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 Xk, k<=6. They are well fitted by the Ansatz Xk = 1/12 k2 + 1/48 k + (1-k)C, with C = 0.0181+-0.0006.
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