Theory of extremal dynamics with quenched disorder: Invasion percolation and related models

The study of phenomena such as capillary displacement in porous media, fracture propagation, and interface dynamics in quenched random media has attracted a great deal of interest in the last few years. This class of problems does not seem to be treatable with the standard theoretical methods, and the only analytical results come from scaling theory or mapping, for some of their properties, to other solvable models. In this paper a recently proposed approach to problems with extremal dynamics in quenched disordered media, named run time statistics (RTS) or quenched-stochastic transformation, is described in detail. This method allows is to map a quenched dynamics such as invasion percolation onto a stochastic annealed process with cognitive memory. By combining RTS with the fixed scale transformation approach, we develop a general and systematic theoretical method to compute analytically the critical exponents of invasion percolation, with and without trapping, and directed invasion percolation. In addition we can also understand and describe quantitatively the self-organized nature of the process.