RANDOM-WALK IN A STRONGLY INHOMOGENEOUS ENVIRONMENT AND INVASION PERCOLATION

Motivated by d-dimensional diffusion in a gradient drift field with sm all diffusion constant epsilon, we consider an inhomogeneous, but reve rsible, continuous time nearest neighbor random walk X(t)(epsilon) on Z(d), or on some other locally finite graph. Let G(n) epsilon be the r andom subgraph whose edges are the first n distinct edges traversed by X(t)(epsilon). We prove that if the strongly inhomogeneous (epsilon - -> 0) limit respects some ordering O of all edges, then (G(0)(epsilon) , G(1)(epsilon), G(2)(epsilon),...) converges to invasion percolation for that O.