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Aspects of first passage percolation. (English) Zbl 0602.60098
École d’été de probabilités de Saint-Flour XIV - 1984, Lect. Notes Math. 1180, 125-264 (1986).
[For the entire collection see Zbl 0579.00013.]
This is largely a survey paper of the advancing boundary of first-passage percolation theory, containing also many new results and insights. Particular emphasis is placed upon the shape problem: What can be said about the asymptotic growth rate of the set of points attainable from the origin in less than time t, as $$t\to \infty ?$$ There are careful discussions of some progress towards a small-deviation theorem for first- passage times, as well as a revised account of the large-deviation theory.
In two dimensions, first-passage percolation is basically equivalent to a problem in flows through randomly-capacitated networks. This suggests the problem of studying the network flow problem in its own right in higher dimensions. Certain results are reported in this direction. The paper terminates with a list of open problems.
Reviewer: G.Grimmett

MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60F10 Large deviations