Models of First-Passage Percolation

First-passage percolation (FPP) was introduced by Hammersley and Welsh in 1965 (see [26]) as a model of fluid flow through a randomly porous material. Envision a fluid injected into the material at a fixed site: as time elapses, the portion of the material that is wet expands in a manner that is a complicated function of the material’s random structure. In the standard FPP model, the spatial randomness of the material is represented by a family of non-negative i.i.d. random variables indexed by the nearest neighbor edges of the Z d lattice. (We take d ≥ 2 throughout this chapter.) If edge e has endpoints u, v ∈ Z d (so |u − v| = 1, where | · | denotes the usual Euclidean norm) then the associated quantity τ(e) represents the time it takes fluid to flow from site u to site v, or the reverse, along the edge e. If the sequence of edges r = (e 1,..., e n ) forms a path from u ∈ Z d to v ∈ Z d , then T(r) ≡ ∑ i τ(e i ) represents the time it takes fluid to flow from u to v along the path r. For any u, v ∈ Z d , we further define the passage time from u to v as $$T\left( {u,v} \right) \equiv \inf \left\{ {T\left( r \right):the\,edgesinrformapathfromutov} \right\}.$$ (1.1)