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**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.

Reviewer: Piotr Garbaczewski (Zielona Góra)

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B43 | Percolation |

### Keywords:

first-passage percolation; large deviations; random variables on graphs; dynamics on graphs
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\textit{Y. Chow} and \textit{Y. Zhang}, Ann. Appl. Probab. 13, No. 4, 1601--1614 (2003; Zbl 1038.60093)

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### References:

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