Local Neighbourhoods for First-Passage Percolation on the Configuration Model

  • Steffen Dereich
  • Marcel Ortgiese


We consider first-passage percolation on the configuration model. Once the network has been generated each edge is assigned an i.i.d. weight modeling the passage time of a message along this edge. Then independently two vertices are chosen uniformly at random, a sender and a recipient, and all edges along the geodesic connecting the two vertices are coloured in red (in the case that both vertices are in the same component). In this article we prove local limit theorems for the coloured graph around the recipient in the spirit of Benjamini and Schramm. We consider the explosive regime, in which case the random distances are of finite order, and the Malthusian regime, in which case the random distances are of logarithmic order.


First passage percolation Random graphs Configuration model Local limit Geodesics Branching processes 

Mathematics Subject Classification

Primary 05C80 Secondary 60J80 


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Authors and Affiliations

  1. 1.Institut für Mathematische StatistikWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK

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