All Spatial Random Graphs with Weak Long-Range Effects have Chemical Distance Comparable to Euclidean Distance

Lukas Lüchtrath ()
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Lukas Lüchtrath: Weierstrass Institute for Applied Analysis and Stochastics

Journal of Theoretical Probability, 2026, vol. 39, issue 1, 1-18

Abstract: 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. We apply the result to various examples, thereby confirming a conjecture on graph distances in the heavy-tailed Boolean model posed by Hirsch (Braz J Probab Stat 31(1):111–143, 2017). The proof is based on a renormalisation scheme introduced by Berger ( arXiv:math/0409021 [math.PR], 2004).

Keywords: Graph distances; Spatial random graphs; Boolean model; Weight-dependent random connection model; Strong decay regime; Polynomial correlations; Long-range percolation; Primary 60K35; Secondary 90B15; 05C80 (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/s10959-025-01467-0

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