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Noise sensitivity in continuum percolation

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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Correspondence to Daniel Ahlberg.

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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

Keywords

  • Boolean Function
  • Percolation Model
  • Poisson Point Process
  • Boolean Model
  • Noise Sensitivity