## Large deviations in first-passage percolation.(English)Zbl 1038.60093

A specific probability space and measure are associated with nonnegative independent identically distributed random variables which are attached to normalized edges of a graph in ($$d>2$$)-dimensional space. Each sample path between two vertices is an alternating sequence of vertices and edges with fixed end-points. Among various passage times, the face-face and point-point first-passage times are of major interest. The paper focuses on their limiting behaviour when the number of edges grows to infinity. Most limit behaviors in first passage percolation are obtained by using subadditive arguments which do not work in the limit. Technical Theorems 1-3 resolve this difficulty by employing the min-cut and max-flow theorems and setting up a multi-subadditive argument.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation
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### References:

 [1] Durrett, R. (1996). Probability : Theory and Examples , 2nd ed. Wadsworth, Belmont, CA. · Zbl 1202.60001 [2] Grimmett, G. and Kesten, H. (1984). First-passage percolation, network flows and electrical resistances. Z. Wahrsch. Verw. Gebiete 66 335–366. · Zbl 0525.60098 [3] Hammersley, J. M. and Welsh, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks and generalized renewal theory. In Bernoulli , Bayes , Laplace Anniversary Volume (J. Neyman and L. Le Cam, eds.) 61–110. Springer, Berlin. · Zbl 0143.40402 [4] Kesten, H. (1986). Aspects of First-Passage Percolation . Lecture Notes in Math. 1180 125–264. Springer, Berlin. · Zbl 0602.60098 [5] Smythe, R. T. and Wierman, J. C. (1978). First Passage Percolation on the Square Lattice . Lecture Notes in Math. 671 . Springer, Berlin. · Zbl 0379.60001 [6] Zhang, Y. (1995). Supercritical behaviors in first-passage percolation. Stochastic Process. Appl. 59 251–266. · Zbl 0840.60090
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