# Limit theorems for maximum flows on a lattice

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## Abstract

We independently assign a non-negative value, as a capacity for the quantity of flows per unit time, with a distribution *F* to each edge on the \(\mathbf{Z}^d\) lattice. We consider the maximum flows through the edges from a source to a sink in a large cube. In this paper, we show that the ratio of the maximum flow and the size of the source is asymptotic to a constant. This constant is denoted by the flow constant. By the max-flow and min-cut theorem, this is equivalent to a statement about the asymptotic behavior of the minimal value assigned to any surface on the large cube. We can also show that there exists such a surface that is proportional to the size of the faces of the cube.

### Keywords

Maximum flow and minimum cut Random surfaces Cluster boundary First passage percolation### Mathematics Subject Classification

60K 35## Notes

### Acknowledgements

The author is grateful to the referee who read the paper carefully, and presented a long report with many detailed and valuable comments and suggestions to improve the exposition. The author would also like to thank Cerf and Theret for their encouragement and many suggestions.

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