Critical Fragmentation Properties of Random Drilling: How Many Holes Need to Be Drilled to Collapse a Wooden Cube?

Abstract

A solid wooden cube fragments into pieces as we sequentially drill holes through it randomly. This seemingly straightforward observation encompasses deep and nontrivial geometrical and probabilistic behavior that is discussed here. Combining numerical simulations and rigorous results, we find off-critical scale-free behavior and a continuous transition at a critical density of holes that significantly differs from classical percolation.

Percolation in Media with Columnar Disorder

“…On the other hand we showed that drilling percolation as treated in [22,23] is very similar, and we argued that the power law behaviors in the subcritical phase found mathematically in [22] are also manifestations of a Griffiths phase. If this is true, we might expect that the extreme anisotropy of critical clusters found in the present paper should also be seen asymptotically in drilling percolation.…”

“…On the other hand, the bound is also the same as in the drilling percolation problem of [22]. The proof uses essentially the arguments, except for the fact that the area A had to be a narrow strip along the diagonal in [22]. This was necessary because only in this way the transversely drilled cylinders correspond to short range disorder within A.…”

“…a square of size × with ≤ L. But the shape of A can be arbitrary, provided it is characterized by length scales much larger than the correlation length ξ(p B ) of 3-d site percolation with p = p B . In particular we can also take a rectangle L × ξ(p B ) or a strip of width ξ(p B ) along one of the two diagonals, as considered in [22]. We assume of course that L ξ(p B ).…”

“…These include percolation in media with long range correlations [19], pacman and interlacements percolation [16][17][18] and 'drilling percolation' [20][21][22][23][24], where the point defects appearing in Bernoulli site percolation are replaced by removed ('drilled') entire columns of sites. But more spectacular is the model of growing random networks of Callaway et al [25] that shows a KosterlitzThouless (KT) transition, and the very old model of 1-d percolation with long range links [26] that shows a KTlike transition that is indeed discontinuous [27].…”

Mechanical Interaction between Cells Facilitates Molecular Transport

“…At a critical fiber occupation fraction of φ 3D C , the continuity of the empty space in the system is completely lost, and the molecule becomes trapped. This exact model has been numerically studied in the context of linear holes being removed from the lattice and has been characterized with the drilling percolation density φ 3D C ≈ 0.75 [70,71]. This 3D percolation problem is nontrivial due to the complicated correlations between the fiber blocks in the linear arrangement of the fibers, and therefore, its effects on diffusion have not, to our knowledge, been studied.…”