First-passage percolation, network flows and electrical resistances

Abstract

We show that the first-passage times of first-passage percolation on ℤ2 are such that P(θ0n n(μ+ɛ)) decay geometrically as n→∞, where θ may represent any of the four usual first-passage-time processes. The former estimate requires no moment condition on the time coordinates, but there exists a geometrically-decaying estimate for the latter quantity if and only if the time coordinate distribution has finite moment generating function near the origin. Here, μ is the time constant and ɛ>0. We study the line-to-line first-passage times and describe applications to the maximum network flow through a randomly-capacitated subsection of ℤ2, and to the asymptotic behaviour of the electrical resistance of a subsection of ℤ2 when the edges of the subsection are wires in an electrical network with random resistances. In the latter case we show, for example, that if each edge-resistance equals 1 or ∞ ohms with probabilities p and 1−p respectively, then the effective resistance Rn across opposite faces of an n by n box satisfies the following: (a) if p<1/2 then P(Rn=∞)→1 as n→∞, (b) if p>1/2 then there exists ν(p)<∞ such that $$P(p^{ - 1} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \mathop {\lim \inf R_n }\limits_{n \to \infty } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \mathop {\lim \sup R_n }\limits_{n \to \infty } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } v(p)) = 1$$ .