First-passage percolation is a time-dependent generalization of discrete Bernoulli percolation in which each graph edge of is assigned a nonnegative random variable called a time coordinate, the collection of which are identically and independently distributed . Within this model, the main objects of study are the asymptotic properties as of the set
(1)
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where
(2)
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is the so-called travel time from to and where
(3)
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is the so-called passage time of a path on which runs successively through the edges . is interpreted as the collection of vertices which can be reached from the origin by time .
Site versions of the first-passage model in which the 's are assigned to sites rather than bonds have also been considered though haven't been written about extensively (Kesten 1987).