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Optimal Manifolds in Random Media

Two methods for the exact calculation of optimal manifolds in random media are presented. The first method is an extension of a transfer operator (TO) technique to nondirected manifolds (“returning paths”). The method is applied to the percolation problems in two dimensions. In the second approach the calculation is mapped onto a random resistor network (RRN) problem. The RRN formulation allows us to find optimal manifolds by the solution of a boundary value problem. Whereas the TO technique is very powerful for the calculation of 1D manifolds the RRN formalism has more potential for higher-dimensional applications.