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.
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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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Zhang, Y. Limit theorems for maximum flows on a lattice. Probab. Theory Relat. Fields 171, 149–202 (2018). https://doi.org/10.1007/s00440-017-0775-z
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DOI: https://doi.org/10.1007/s00440-017-0775-z