We consider random labelings of finite graphs conditioned on a small fixed number of peaks. We introduce a continuum framework where a combinatorial graph is associated with a metric graph and edges are identified with intervals. Next we consider a sequence of partitions of the edges of the metric graph with the partition size going to zero. As the mesh of the subdivision goes to zero, the conditioned random labelings converge, in a suitable sense, to a deterministic function which evolves as an increasing process of subsets of the metric graph that grows at rate one while maximizing an appropriate notion of entropy. We call such functions floodings. We present a number of qualitative and quantitative properties of floodings and some explicit examples.

中文翻译：

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度量图的泛滥

我们考虑以固定数量的小峰为条件的有限图的随机标记。我们引入了一个连续统框架，其中组合图与度量图相关联，边用间隔标识。接下来，我们考虑度量图边缘的一系列分区，分区大小为零。随着细分的网格变为零，条件随机标记在适当的意义上收敛到确定性函数，该函数随着度量图的子集的增加过程而演变，该过程以速率 1 增长，同时最大化适当的熵概念。我们称这些函数为泛洪。我们介绍了洪水的许多定性和定量特性以及一些明确的例子。