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Cycle structure of percolation on high-dimensional tori

Remco van der HofstadArtëm Sapozhnikov — 2014

Annales de l'I.H.P. Probabilités et statistiques

In the past years, many properties of the largest connected components of critical percolation on the high-dimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one for percolation on the complete graph or Erdős–Rényi random graph, raising the question whether the scaling limits of the largest connected components, as identified by Aldous (1997), are also equal. In this paper, we investigate the of the largest critical components...

Scale-free percolation

Maria DeijfenRemco van der HofstadGerard Hooghiemstra — 2013

Annales de l'I.H.P. Probabilités et statistiques

We formulate and study a model for inhomogeneous long-range percolation on d . Each vertex x d is assigned a non-negative weight W x , where ( W x ) x d are i.i.d. random variables. Conditionally on the weights, and given two parameters α , λ g t ; 0 , the edges are independent and the probability that there is an edge between x and y is given by p x y = 1 - exp { - λ W x W y / | x - y | α } . The parameter λ is the percolation parameter, while α describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there...

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