Critical path analysis of conductance fluid invasion and rupture

Critical path analysis (CPA) is a very useful tool in treating transport properties of strongly heterogeneous conducting materials. In this work we show how the range of applicability of CPA is extended to percolating behavior induced by diverse dynamical processes. We illustrate this through useful applications in the context of conductance of random resistor networks, fluid invasion of a porous matrix and tough rupture of disordered materials. For random conducting systems it is reviewed that there is a formal analogy between transport on fractal (percolating) structures and transport in strongly heterogeneous networks. For invasion percolation, we show that CPA holds exactly for all dimensions and distributions of disorder. Finally, for toughness tests it is shown that the strength of an undiluted system is identical to that of the toughest directed percolation backbone. A common thread is laid among these problems by posing them in terms of scale-invariant relations of system properties on hierarchical lattices.