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Local Neighbourhoods for First-Passage Percolation on the Configuration Model

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Abstract

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.

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Authors and Affiliations

  1. Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster, Einsteinstraße 62, 48149, Münster, Germany

    Steffen Dereich

  2. Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK

    Marcel Ortgiese

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  1. Steffen Dereich
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  2. Marcel Ortgiese
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Correspondence to Steffen Dereich.

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Dereich, S., Ortgiese, M. Local Neighbourhoods for First-Passage Percolation on the Configuration Model. J Stat Phys 173, 485–501 (2018). https://doi.org/10.1007/s10955-018-2028-7

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  • DOI: https://doi.org/10.1007/s10955-018-2028-7

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