Probability Theory and Related Fields

, Volume 171, Issue 1–2, pp 149–202 | Cite as

Limit theorems for maximum flows on a lattice



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.


Maximum flow and minimum cut Random surfaces Cluster boundary First passage percolation 

Mathematics Subject Classification

60K 35 



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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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ColoradoColorado SpringsUSA

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