Random walk in a strongly inhomogeneous environment and invasion percolation (Q1346404)

The invasion percolation is a young branch of percolation theory. In contrast to the standard Bernoulli percolation, which in general describes the passive diffusion of fluid particles, the invasion percolation reflects properties of active displacement (for example, when one fluid forces out another one under pressure). It is shown that the invasion percolation can arise resulting from the strongly inhomogeneous limit \(\varepsilon \to 0\) of the diffusion process \[ dY_ t = \sqrt \varepsilon dB_ t - {1 \over 2} \nabla W (Y_ t)dt, \] where \(B_ t\) is the standard Wiener process and \(W\) is a (nonrandom) smooth function with countable set of local minimum points. If \(Y_ t\) is observed unfrequently [\textit{C. Kipnis} and \textit{C. M. Newman}, SIAM J. Appl. Math. 45, 972-982 (1985; Zbl 0592.60063)], it should have essentially the same asymptotic \(\varepsilon \to 0\) behaviour as a Markov jump process \(X^ \varepsilon_ t\) with the state space \(\Lambda\) consisting of points of local minima of \(W\). It is proved that if the waiting times of \(X^ \varepsilon_ t\) attribute some ordering \(\Theta\) to the graph \(\Gamma\) with the vertex set \(\Delta\) and edges between neighbouring pairs, then the sequence of transitions made for the first time coincides in the limit \(\varepsilon \to 0\) with the invasion percolation with respect to \(\Theta\). The proof is based on two useful statements concerning properties of random walks and invasion percolation, respectively.