Electronic Journal of Probability

Euclidean partitions optimizing noise stability

Steven Heilman

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The Standard Simplex Conjecture of Isaksson and Mossel asks for the partition $\{A_{i}\}_{i=1}^{k}$ of $\mathbb{R}^{n}$ into $k\leq n+1$ pieces of equal Gaussian measure of optimal noise stability.  That is, for $\rho>0$, we maximize$$\sum_{i=1}^{k}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}1_{A_{i}}(x)1_{A_{i}}(x\rho+y\sqrt{1-\rho^{2}})e^{-(x_{1}^{2}+\cdots+x_{n}^{2})/2}e^{-(y_{1}^{2}+\cdots+y_{n}^{2})/2}dxdy.$$Isaksson and Mossel guessed the best partition for this problem and proved some applications of their conjecture. For example, the Standard Simplex Conjecture implies the Plurality is Stablest Conjecture. For $k=3,n\geq2$ and $0<\rho<\rho_{0}(k,n)$, we prove the Standard Simplex Conjecture. The full conjecture has applications to theoretical computer science and to geometric multi-bubble problems (after Isaksson and Mossel).

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 71, 37 pp.

Accepted: 15 August 2014
First available in Project Euclid: 4 June 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 68Q25: Analysis of algorithms and problem complexity [See also 68W40]

Standard simplex plurality optimization MAX-k-CUT Unique Games Conjecture

This work is licensed under a Creative Commons Attribution 3.0 License.


Heilman, Steven. Euclidean partitions optimizing noise stability. Electron. J. Probab. 19 (2014), paper no. 71, 37 pp. doi:10.1214/EJP.v19-3083. https://projecteuclid.org/euclid.ejp/1465065713

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