Continuity of the percolation probability and chemical distances in inhomogeneous long-range percolation

Hazra, R. ; Wüthrich, M. (2014) Continuity of the percolation probability and chemical distances in inhomogeneous long-range percolation arXiv: Probability . pp. 1-24.

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Related URL: http://dx.doi.org/arXiv preprint arXiv:1401.0409

Abstract

Inhomogeneous long-range percolation on the lattice Zd was introduced in [12] as an extension of the homogeneous long-range percolation model. The inhomogeneous long-range percolation model assigns iid weights Wx to each vertex x∈ Zd. Conditionally on these weights, an edge between vertices x and y is occupied with probability pxy= 1− exp (− λWxWy| x− y|− α), independently of all other edges.[12] provides the phase transition picture for the existence of an infinite component of occupied edges. In the present paper we complement this phase transition picture by proving that the percolation probability (as a function of λ) is continuous for α∈(d, 2d) and, therefore, there is no infinite component at criticality. Moreover, we complement the picture of [12] about chemical distances in the infinite component.

Item Type:Article
Source:Copyright of this article belongs to author(s).
Keywords:Long Range Percolation; Scale Free Percolation; Chemical Distance; Continuity; Inhomogeneous Long Range Percolation.
ID Code:121111
Deposited On:09 Jul 2021 09:15
Last Modified:09 Jul 2021 09:15

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