Monochromatic path crossing exponents and graph connectivity in 2D percolation

TL;DR

This paper investigates the fractal dimensions of k-connected parts of 2D percolation clusters, deriving new numerical estimates for crossing exponents that describe the probability decay of multiple disjoint paths crossing annuli.

Contribution

It introduces a transfer matrix method to numerically compute the crossing exponents X_k for k up to 6, generalizing known cluster and backbone dimensions.

Findings

Derived a fitting formula for X_k with high accuracy

Provided numerical estimates for X_k for k ≤ 6

Enhanced understanding of multi-path crossing probabilities in 2D percolation

Abstract

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) ~ (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<=6. They are well fitted by the Ansatz X_k = 1/12 k^2 + 1/48 k + (1-k)C, with C = 0.0181+-0.0006.