HAL CCSD
Shape of shortest paths in random spatial networks
Kartun-Giles, Alexander
Barthelemy, Marc
Dettmann, Carl
Max Planck Institute for Mathematics in the Sciences (MPI-MiS) ; Max-Planck-Gesellschaft
Institut de Physique Théorique - UMR CNRS 3681 (IPHT) ; Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)
School of Mathematics [Bristol] ; University of Bristol [Bristol]
International audience
ISSN: 2470-0045
EISSN: 2470-0053
Physical Review E
American Physical Society (APS)
cea-02564008
https://cea.hal.science/cea-02564008
https://cea.hal.science/cea-02564008/document
https://cea.hal.science/cea-02564008/file/Barth1.pdf
https://cea.hal.science/cea-02564008
Physical Review E , 2019, 100 (3), pp.2315. ⟨10.1103/PhysRevE.100.032315⟩
DOI: 10.1103/PhysRevE.100.032315
info:eu-repo/semantics/altIdentifier/doi/10.1103/PhysRevE.100.032315
en
[PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech]
info:eu-repo/semantics/article
Journal articles
In the classic model of first passage percolation, for pairs of vertices separated by a Euclidean distance L, geodesics exhibit deviations from their mean length L that are of order L$^χ$ , while the transversal fluctuations, known as wandering, grow as L$^ξ$. We find that when weighting edges directly with their Euclidean span in various spatial network models, we have two distinct classes defined by different exponents ξ = 3/5 and χ = 1/5, or ξ = 7/10 and χ = 2/5, depending only on coarse details of the specific connectivity laws used. Also, the travel time fluctuations are Gaussian, rather than Tracy-Widom, which is rarely seen in first passage models. The first class contains proximity graphs such as the hard and soft random geometric graph, and the k-nearest neighbour random geometric graphs, where via Monte Carlo simulations we find ξ = 0.60 ± 0.01 and χ = 0.20 ± 0.01, showing a theoretical minimal wandering. The second class contains graphs based on excluded regions such as β-skeletons and the Delaunay triangulation and are characterized by the values ξ = 0.70 ± 0.01 and χ = 0.40 ± 0.01, with a nearly theoretically maximal wandering exponent. We also show numerically that the KPZ relation χ = 2ξ − 1 is satisfied for all these models. These results shed some light on the Euclidean first passage process, but also raise some theoretical questions about the scaling laws and the derivation of the exponent values, and also whether a model can be constructed with maximal wandering, or non-Gaussian travel fluctuations, while embedded in space.
2019-09
info:eu-repo/semantics/OpenAccess