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Limiting shape for first-passage percolation models on random geometric graphs

Part of: Special processes Limit theorems General convexity

Published online by Cambridge University Press:  24 April 2023

Cristian F. Coletti ,
Lucas R. de Lima Open the ORCID record for Lucas R. de Lima [Opens in a new window] ,
Alexander Hinsen ,
Benedikt Jahnel  and
Daniel Valesin
Cristian F. Coletti*
Affiliation:
Federal University of ABC (UFABC)
Lucas R. de Lima*
Affiliation:
Federal University of ABC (UFABC)
Alexander Hinsen*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
Benedikt Jahnel*
Affiliation:
Technische Universität Braunschweig
Daniel Valesin*
Affiliation:
University of Warwick
*
*Postal address: Center for Mathematics, Computation, and Cognition, Universidade Federal do ABC, Avenida dos Estados, 5001 Bangu, Santo André, São Paulo, Brazil.
*Postal address: Center for Mathematics, Computation, and Cognition, Universidade Federal do ABC, Avenida dos Estados, 5001 Bangu, Santo André, São Paulo, Brazil.
****Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany. Email: alexander.hinsen@wias-berlin.de
*****Postal address: Institut für Mathematische Stochastik, Technische Universität Braunschweig, Universitätsplatz 2, 38106 Braunschweig, Germany & Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany. Email: benedikt.jahnel@tu-braunschweig.de
******Postal address: Department of Statistics, University of Warwick, Coventry, CV4 7AL, United Kingdom. Email: daniel.valesin@warwick.ac.uk
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Abstract

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape, and we show that the shape is a Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. In the latter case we further show that it converges weakly to a nonstandard branching process in the joint limit of large intensities and slow passage times.

Keywords

Poisson–Gilbert graphRichardson modelBernoulli bond percolationhigh-density limitisotropic shapes

MSC classification

Primary: 52A22: Random convex sets and integral geometry 60F15: Strong theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 4 , December 2023 , pp. 1367 - 1385
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Auffinger, A., Damron, M. and Hanson, J. (2017). 50 Years of First-Passage Percolation (University Lecture Ser. 68). American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Deijfen, M. (2003). Asymptotic shape in a continuum growth model. Adv. Appl. Prob. 35, 303–318.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (2009). Markov Processes: Characterization and Convergence. Wiley, Chichester.Google Scholar
Hammersley, J. and Welsh, D. (1965). First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Bernoulli 1713 Bayes 1763 Laplace 1813, eds. J. Neyman and L. M. Cam. Springer, New York, pp. 61–110.Google Scholar
Hirsch, C., Neuhäuser, D., Gloaguen, C. and Schmidt, V. (2015). First passage percolation on random geometric graphs and an application to shortest-path trees. Adv. Appl. Prob. 47, 328–354.CrossRefGoogle Scholar
Jahnel, B. and König, W. (2020). Probabilistic Methods In Telecommunications. Birkhäuser, Basel.CrossRefGoogle Scholar
Kesten, H. (1986). Aspects of first passage percolation. In École d’été de probabilités de Saint Flour, 1984 (Lect. Notes Math. 1180), eds. R. Carmona, H. Kesten and J. B. Walsh. Springer, Berlin, pp. 125–264.Google Scholar
Kingman, J. (1968). The ergodic theory of subadditive stochastic processes. J. R. Statist. Soc. B 30, 499–510.Google Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation (Cambridge Tracts Math. 119). Cambridge University Press.Google Scholar
Ménard, L. and Singh, A. (2016). Percolation by cumulative merging and phase transition for the contact process on random graphs. Ann. Sci. École Normale Supérieure (4) 49, 1189–1238.CrossRefGoogle Scholar
Penrose, M. (2003). Random Geometric Graphs (Oxford Studies Prob. 5). Oxford University Press.Google Scholar
Penrose, M. and Pisztora, A. (1996). Large deviations for discrete and continuous percolation. Adv. Appl. Prob. 28, 29–52.CrossRefGoogle Scholar
Prokhorov, Y. V. (1956). Convergence of random processes and limit theorems in probability theory. Theory Prob. Appl. 1, 157–214.CrossRefGoogle Scholar
Riblet, T. (2019). Le processus de contact sur le graphe booléen. PhD thesis, Université de Lorraine.Google Scholar
Richardson, D. (1973). Random growth in a tessellation. Proc. Camb. Phil. Soc. 74, 515–528.CrossRefGoogle Scholar
Rosenthal, H. (1970). On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables. Israel J. Math. 8, 273–303.Google Scholar
Torquato, S. and Jiao, Y. (2012). Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses. J. Chem. Phys. 137, 074106.CrossRefGoogle ScholarPubMed
Yao, C.-L., Chen, G. and Guo, T.-D. (2011). Large deviations for the graph distance in supercritical continuum percolation. J. Appl. Prob. 48, 154–172.CrossRefGoogle Scholar