Sharp phase transition for Cox percolation

Hirsch, Christian R; Jahnel, Benedikt; Muirhead, Stephen

doi:10.48550/arxiv.2203.01251

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2022

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“…The existence of long-range dependencies has serious consequences for percolation in the Cox-Boolean model based on Manhattan grids. For example, there is no sharp-threshold phenomenon [AB87, DCT16,HJM22]. More precisely, in the subcritical percolation regime, the probability of the event that the cluster of the origin reaches beyond a large ball does not decrease exponentially, see Proposition 5.…”

Section: Introductionmentioning

confidence: 99%

Continuum Percolation in a Nonstabilizing Environment

Jahnel¹,

Jhawar²,

Vu³

2022

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We prove nontrivial phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson-Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox-Boolean model. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in [Hof05].

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Section: Introductionmentioning

confidence: 99%