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Maximally stable Gaussian partitions with discrete applications

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Abstract

Gaussian noise stability results have recently played an important role in proving results in hardness of approximation in computer science and in the study of voting schemes in social choice. We prove a new Gaussian noise stability result generalizing an isoperimetric result by Borell on the heat kernel and derive as applications:

  • An optimality result for majority in the context of Condorcet voting.

  • A proof of a conjecture on “cosmic coin tossing” for low influence functions.

We also discuss a Gaussian noise stability conjecture which may be viewed as a generalization of the “Double Bubble” theorem and show that it implies:

  • A proof of the “Plurality is Stablest Conjecture”.

  • That the Frieze-Jerrum SDP for MAX-q-CUT achieves the optimal approximation factor assuming the Unique Games Conjecture.

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Authors and Affiliations

  1. Chalmers University of Technology and Göteborg University, SE-41296, Göteborg, Sweden

    Marcus Isaksson

  2. Department of Statistics, University of California, 423 Evans Hall, Berkeley, CA, 94720-3860, USA

    Elchanan Mossel

  3. Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Elchanan Mossel

Authors
  1. Marcus Isaksson
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  2. Elchanan Mossel
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Correspondence to Marcus Isaksson.

Additional information

Supported by an Alfred Sloan fellowship in Mathematics, by NSF CAREER grant DMS-0548249 (CAREER), by DOD ONR grant (N0014-07-1-05-06), by BSF grant 2004105, by ISF grant 1300/08, by a Minerva Foundation grant and by ERC Marie Curie Grant 2008 239317.

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Isaksson, M., Mossel, E. Maximally stable Gaussian partitions with discrete applications. Isr. J. Math. 189, 347–396 (2012). https://doi.org/10.1007/s11856-011-0181-7

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  • DOI: https://doi.org/10.1007/s11856-011-0181-7

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