The fractal dimension of the minimum path in two - and three - dimensional percolation

We calculate the fractal dimension d, , , of the shortest path I between two points on a percolation cluster, where 1 rd”n and r is the Pythagorean distance between the points. We find d , , ,= 1.130*0.002 for d = 2 and 1.34i0.01 for d =3 . What is the length 1 of the shortest path or ‘chemical distance’ between two points of a random material? In general, I is greater than r, the Pythagorean distance between the points. If the object is self-similar (‘fractal’) on length scales r < 6 (where 6 is the pair connectedness length), then by definition the density p decreases as the size increases as p r d f d , where dr is the fractal dimension. That dfd is negative implies that when a fractal is examined on larger and larger scales there must occur ‘holes’ on larger and larger scales, up to the size of the connectedness length 6. Now as r increases, 1 increases faster since larger and larger ‘holes’ must be circumnavigated by the shortest path. Previous work suggests that