The Power Laws of Geodesics in Some Random Sets with Dilute Concentration of Inclusions

摘要

A method for computing upper-bounds on the length of geodesics spanning random sets in 2D and 3D is proposed, with emphasis on Boolean models containing a vanishingly small surface or volume fraction of inclusions f 1. The distance function is zero inside the grains and equal to the Euclidean distance outside of them, and the geodesics are shortest paths connecting two points far from each other. The asymptotic behavior of the upper-bounds is derived in the limit f → 0. The scalings involve powerlaws with fractional exponents ～f~(2/3) for Boolean sets of disks or aligned squares and ～f~(1/2) for the Boolean set of spheres. These results are extended to models of hyperspheres in arbitrary dimension and, in 2D and 3D, to a more general problem where the distance function is non-zero in the inclusions. Finally, other fractional exponents are derived for the geodesics spanning multiscale Boolean sets, based on inhomogeneous Poisson point processes, in 2D and 3D.