2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/223855In the 1980s an important goal of the emergent field of fractals was to determine the relationships between their physical and geometrical properties. The fractal-Einstein and Alexander-Orbach laws, which interrelate electrical, diffusive and fractal properties, are two key theories of this type. Here we settle a long standing controversy about their exactness by showing that the properties of a class of fractal trees violate both laws. A new formula is derived which unifies the two classical results by proving that if one holds, then so must the other, and resolves a puzzling discrepancy in the properties of Eden trees and diffusion limited aggregates. The failure of the classical laws is attributed to anisotropic exploration of the network by a random walker. The occurrence of this newly revealed behaviour means that numerous theories, such as recent first passage time results, are restricted to a narrower range of networks than previously thought.Statistical Mechanicstext