SummaryWe show that the first-passage times of first-passage percolation on ℤ2 are such that P(θ0nn(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
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$$P(p^{ - 1} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \mathop {\lim \inf R_n }\limits_{n \to \infty } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \mathop {\lim \sup R_n }\limits_{n \to \infty } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } v(p)) = 1$$
.
There are some corresponding results for certain other two-dimensional lattices, and for higher dimensions.
[1]
J. Straley.
Critical exponents for the conductivity of random resistor lattices
,
1977
.
[2]
Harry Kesten,et al.
On the time constant and path length of first-passage percolation
,
1980,
Advances in Applied Probability.
[3]
S. Kirkpatrick.
Models of Disordered Materials
,
1983
.
[4]
Critical sponge dimensions in percolation theory
,
1981,
Advances in Applied Probability.
[5]
H. Kesten.
The critical probability of bond percolation on the square lattice equals 1/2
,
1980
.
[6]
J. T. Cox,et al.
Some Limit Theorems for Percolation Processes with Necessary and Sufficient Conditions
,
1981
.
[7]
D. Stauffer.
Scaling Theory of Percolation Clusters
,
1979,
Complex Media and Percolation Theory.
[8]
G. Grimmett,et al.
Flow in networks with random capacities
,
1982
.
[9]
R. Smythe,et al.
First-passage percolation on the square lattice. I
,
1977,
Advances in Applied Probability.
[10]
H. Kesten.
Percolation theory for mathematicians
,
1982
.
[11]
George Papanicolaou,et al.
Bounds for effective parameters of heterogeneous media by analytic continuation
,
1983
.
[12]
Rolf Künnemann,et al.
The diffusion limit for reversible jump processes onZd with ergodic random bond conductivities
,
1983
.
[13]
B. Gnedenko,et al.
Limit Distributions for Sums of Independent Random Variables
,
1955
.
[14]
J. Hammersley,et al.
First-Passage Percolation, Subadditive Processes, Stochastic Networks, and Generalized Renewal Theory
,
1965
.
[15]
On the continuity of the time constant of first-passage percolation
,
1981
.