WIAS Preprint No. 3154, (2024)

All spatial graphs with weak long-range effects have chemical distance comparable to Euclidean distance


Authors

  • Lüchtrath, Lukas
    ORCID: 0000-0003-4969-806X

2020 Mathematics Subject Classification

  • 60K35 90B15 05C80

Keywords

  • Graph distances, spatial random graphs, Boolean model, weight-dependent random connection model, strong decay regime, polynomial correlations, long-range percolation

DOI

10.20347/WIAS.PREPRINT.3154

Abstract

This note provides a sufficient condition for linear lower bounds on chemical distances (compared to the Euclidean distance) in general spatial random graphs. The condition is based on the scarceness of long edges in the graph and weak correlations at large distances and is valid for all translation invariant and locally finite graphs that fulfil these conditions. The proof is based on a renormalisation scheme introduced by Berger [arXiv: 0409021 (2004)].

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