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Standard simplices and pluralities are not the most noise stable

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Abstract

The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are two conjectures stating that certain partitions are optimal with respect to Gaussian and discrete noise stability respectively. These two conjectures are natural generalizations of the Gaussian noise stability result by Borell (1985) and the Majority is Stablest Theorem (2004). Here we show that the standard simplex is not the most stable partition in Gaussian space and that Plurality is not the most stable low influence partition in discrete space for every number of parts k ≥ 3, for every value ρ ≠ 0 of the noise and for every prescribed measure for the different parts as long as they are not all equal to 1/k. Our results do not contradict the original statements of the Plurality is Stablest and Standard Simplex Conjectures in their original statements concerning partitions to sets of equal measure. However, they indicate that if these conjectures are true, their veracity and their proofs will crucially rely on assuming that the sets are of equal measures, in stark contrast to Borell’s result, the Majority is Stablest Theorem and many other results in isoperimetric theory. Given our results it is natural to ask for (conjectured) partitions achieving the optimum noise stability.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, CA, 90095-1555, USA

    Steven Heilman

  2. Departments of Statistics and Computer Science, University of Pennsylvania, Philadelphia, PA, 19104, USA

    Elchanan Mossel

  3. University of California, Berkeley, 110 Sproul Hall, Berkeley, CA, 94720-5800, USA

    Elchanan Mossel

  4. Mathematics and Electrical and Computer Engineering, University of Texas at Austin, Austin, TX, 78712, USA

    Joe Neeman

Authors
  1. Steven Heilman
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  2. Elchanan Mossel
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  3. Joe Neeman
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Correspondence to Steven Heilman.

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Heilman, S., Mossel, E. & Neeman, J. Standard simplices and pluralities are not the most noise stable. Isr. J. Math. 213, 33–53 (2016). https://doi.org/10.1007/s11856-016-1320-y

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  • DOI: https://doi.org/10.1007/s11856-016-1320-y

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