Monochromatic path crossing exponents and graph connectivity in 2D percolation

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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.