Euclidean and chemical distances in ellipses percolation

Hilário, Marcelo; Ungaretti, Daniel

doi:10.48550/arxiv.2103.09786

Cited by 3 publications

(3 citation statements)

References 15 publications

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“…The random graph is then constructed by taking the Poisson process as the vertex set and forming edges given the collection of random ellipses between pairs of points of the point process if their ellipses intersect. Hilário and Ungaretti [22] show that, for γ ∈ (1, 2), the model is ultrasmall.…”

Section: The Reinforced Age-dependent Random Connection Modelmentioning

confidence: 99%

Chemical Distance in Geometric Random Graphs with Long Edges and Scale-Free Degree Distribution

Gracar

Grauer

Mörters

2022

Commun. Math. Phys.

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We study geometric random graphs defined on the points of a Poisson process in d-dimensional space, which additionally carry independent random marks. Edges are established at random using the marks of the endpoints and the distance between points in a flexible way. Our framework includes the soft Boolean model (where marks play the role of radii of balls centered in the vertices), a version of spatial preferential attachment (where marks play the role of birth times), and a whole range of other graph models with scale-free degree distributions and edges spanning large distances. In this versatile framework we give sharp criteria for absence of ultrasmallness of the graphs and in the ultrasmall regime establish a limit theorem for the chemical distance of two points. Other than in the mean-field scale-free network models the boundary of the ultrasmall regime depends not only on the power-law exponent of the degree distribution but also on the spatial embedding of the graph, quantified by the rate of decay of the probability of an edge connecting typical points in terms of their spatial distance.

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Section: The Reinforced Age-dependent Random Connection Modelmentioning

confidence: 99%

Chemical Distance in Geometric Random Graphs with Long Edges and Scale-Free Degree Distribution

Gracar

Grauer

Mörters

2022

Commun. Math. Phys.

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“…Note that more subtle results about the percolation of the covered set in the α = 2 case (which arises from the intersection of the 3 dimensional Poisson cylinder model [26] with a plane) and the percolation of the vacant set are also proved in [25]. The geometry of the infinite cluster of ellipses percolation is further studied in [16] in the case when 1 < α < 2.…”

Section: Ellipses Percolationmentioning

confidence: 99%

Percolation of worms

Ráth¹,

Rokob²

2021

Preprint

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We introduce a new correlated percolation model on the d-dimensional lattice Z d called the random length worms model. Assume given a probability distribution on the set of positive integers (the length distribution) and v ∈ (0, ∞) (the intensity parameter). From each site of Z d we start POI(v) independent simple random walks with this length distribution. We investigate the connectivity properties of the set S v of sites visited by this cloud of random walks. It is easy to show that if the second moment of the length distribution is finite then S v undergoes a percolation phase transition as v varies. Our main contribution is a sufficient condition on the length distribution which guarantees that S v percolates for all v > 0 if d ≥ 5. E.g., if the probability mass function of the length distribution isfor some ℓ 0 > e e and ε > 0 then S v percolates for all v > 0. Note that the second moment of this length distribution is only "barely" infinite. In order to put our result in the context of earlier results about similar models (e.g., finitary random interlacements, loop percolation, Poisson Boolean model, ellipses percolation, etc.), we define a natural family of percolation models called the Poisson zoo and argue that the percolative behaviour of the random length worms model is quite close to being "extremal" in this family of models.

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“…The random graph is then constructed by taking the Poisson process as the vertex set and forming edges given the collection of random ellipses between pairs of points of the point process if their ellipses intersect. Hilário and Ungaretti [20] show that, for γ ∈ (1, 2), the model is ultrasmall.…”

Section: Ellipses Percolationmentioning

confidence: 99%

Chemical distance in geometric random graphs with long edges and scale-free degree distribution

Gracar,

Grauer,

Mörters

2021

Preprint

View full text Add to dashboard Cite

We study geometric random graphs defined on the points of a Poisson process in d-dimensional space, which additionally carry independent random marks. Edges are established at random using the marks of the endpoints and the distance between points in a flexible way. Our framework includes the soft Boolean model (where marks play the role of radii of balls centred in the vertices), a version of spatial preferential attachment (where marks play the role of birth times), and a whole range of other graph models with scale-free degree distributions and edges spanning large distances. In this versatile framework we give sharp criteria for absence of ultrasmallness of the graphs and in the ultrasmall regime establish a limit theorem for the chemical distance of two points. Other than in the mean-field scale-free network models the boundary of the ultrasmall regime depends not only on the power-law exponent of the degree distribution but also on the spatial embedding of the graph, quantified by the rate of decay of the probability of an edge connecting typical points in terms of their spatial distance.

show abstract

Euclidean and chemical distances in ellipses percolation

Cited by 3 publications

References 15 publications

Chemical Distance in Geometric Random Graphs with Long Edges and Scale-Free Degree Distribution

Chemical Distance in Geometric Random Graphs with Long Edges and Scale-Free Degree Distribution

Percolation of worms

Chemical distance in geometric random graphs with long edges and scale-free degree distribution

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