Maximally stable Gaussian partitions with discrete applications

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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Correspondence to Marcus Isaksson.

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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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Keywords

  • Social Choice
  • Invariance Principle
  • Social Choice Function
  • Independent Sequence
  • Double Bubble