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Exclusion sensitivity of Boolean functions
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  • Published: 28 January 2012

Exclusion sensitivity of Boolean functions

  • Erik I. Broman1,2,
  • Christophe Garban3 &
  • Jeffrey E. Steif1,2 

Probability Theory and Related Fields volume 155, pages 621–663 (2013)Cite this article

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  • 8 Citations

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Abstract

Recently the study of noise sensitivity and noise stability of Boolean functions has received considerable attention. The purpose of this paper is to extend these notions in a natural way to a different class of perturbations, namely those arising from running the symmetric exclusion process for a short amount of time. In this study, the case of monotone Boolean functions will turn out to be of particular interest. We show that for this class of functions, ordinary noise sensitivity and noise sensitivity with respect to the complete graph exclusion process are equivalent. We also show this equivalence with respect to stability. After obtaining these fairly general results, we study “exclusion sensitivity” of critical percolation in more detail with respect to medium-range dynamics. The exclusion dynamics, due to its conservative nature, is in some sense more physical than the classical i.i.d. dynamics. Interestingly, we will see that in order to obtain a precise understanding of the exclusion sensitivity of percolation, we will need to describe how typical spectral sets of percolation diffuse under the underlying exclusion process.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden

    Erik I. Broman & Jeffrey E. Steif

  2. Department of Mathematical Sciences, Göteborg University, 41296, Gothenburg, Sweden

    Erik I. Broman & Jeffrey E. Steif

  3. CNRS, Ecole Normale Supérieure de Lyon, UMPA, 46 allée d’Italie, 69364, Lyon Cedex 07, France

    Christophe Garban

Authors
  1. Erik I. Broman
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  2. Christophe Garban
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  3. Jeffrey E. Steif
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Correspondence to Jeffrey E. Steif.

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Cite this article

Broman, E.I., Garban, C. & Steif, J.E. Exclusion sensitivity of Boolean functions. Probab. Theory Relat. Fields 155, 621–663 (2013). https://doi.org/10.1007/s00440-011-0409-9

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  • Received: 27 January 2011

  • Revised: 13 December 2011

  • Published: 28 January 2012

  • Issue Date: April 2013

  • DOI: https://doi.org/10.1007/s00440-011-0409-9

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Keywords

  • Noise sensitivity
  • Exclusion sensitivity

Mathematics Subject Classification (2010)

  • 60K35
  • 42B05
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