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Noise sensitivity in continuum percolation

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Abstract

We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first which involves a percolation model with critical probability p c ≠ 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with p c bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.

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Authors and Affiliations

  1. Mathematical Sciences, Chalmers University of Technology, SE-41296, Göteborg, Sweeden

    Daniel Ahlberg & Erik Broman

  2. Mathematical Sciences, University of Gothenburg, Göteborg, Sweden

    Daniel Ahlberg & Erik Broman

  3. IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, Brasil

    Simon Griffiths & Robert Morris

Authors
  1. Daniel Ahlberg
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  2. Erik Broman
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  3. Simon Griffiths
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  4. Robert Morris
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Correspondence to Daniel Ahlberg.

Additional information

Research supported in part by: (DA) The Royal Swedish Academy of Sciences; (EB) The Göran Gustafsson Foundation for Research in Natural Sciences and Medicine; (SG) CNPq bolsa PDJ; (RM) CNPq bolsa de Produtividade em Pesquisa.

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Ahlberg, D., Broman, E., Griffiths, S. et al. Noise sensitivity in continuum percolation. Isr. J. Math. 201, 847–899 (2014). https://doi.org/10.1007/s11856-014-1038-y

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  • DOI: https://doi.org/10.1007/s11856-014-1038-y

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