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First-Passage Percolation

المؤلف:  Grimmett, G

المصدر:  Percolation, 2nd ed. Berlin: Springer-Verlag, 1999.

الجزء والصفحة:  ...

15-5-2022

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First-Passage Percolation

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

{v in Z^d:T(0,v)<=t} " src="https://mathworld.wolfram.com/images/equations/First-PassagePercolation/NumberedEquation1.svg" style="height:25px; width:193px" />

(1)

where

{T(r):r is a path from u to v} " src="https://mathworld.wolfram.com/images/equations/First-PassagePercolation/NumberedEquation2.svg" style="height:21px; width:287px" />

(2)

is the so-called travel time from  to  and where

(3)

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).

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REFERENCES

Grimmett, G. Percolation, 2nd ed. Berlin: Springer-Verlag, 1999.

Kesten, H. "The 1986 Wald Memorial Lectures: Percolation Theory and First-Passage Percolation." Ann. Prob. 15, 1231-1271, 1987.