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

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...

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Bibliographic Details
Published inJournal of applied probability Vol. 60; no. 4; pp. 1367 - 1385
Main Authors Coletti, Cristian F., de Lima, Lucas R., Hinsen, Alexander, Jahnel, Benedikt, Valesin, Daniel
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.12.2023
Subjects
Euclidean geometry
Euclidean space
Growth models
Infections
Original Article
Percolation
Random variables
Poisson–Gilbert graph
Bernoulli bond percolation
isotropic shapes
52A22
high-density limit
60K35
Richardson model
60F15
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Summary: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.
Bibliography:ObjectType-Article-1
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content type line 14
ISSN:0021-9002
1475-6072
DOI:10.1017/jpr.2023.5