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Longest Path Distance in Random Circuits

We study distance properties of a general class of random directed acyclic graphs (dags). In a dag, many natural notions of distance are possible, for there exist multiple paths between pairs of nodes. The distance of interest for circuits is the maximum length of a path between two nodes. We give laws of large numbers for the typical depth (distance to the root) and the minimum depth in a random dag. This completes the study of natural distances in random dags initiated (in the uniform case) by Devroye and Janson. We also obtain large deviation bounds for the minimum of a branching random walk with constant branching, which can be seen as a simplified version of our main result.