Standard Simplices and Pluralities are Not the Most Noise Stable

TL;DR

This paper demonstrates that the standard simplex and plurality partitions are not the most noise-stable in Gaussian and discrete spaces unless all parts are of equal measure, challenging existing conjectures.

Contribution

It shows that the Standard Simplex and Plurality are not optimal for noise stability unless measures are equal, highlighting the importance of equal measure assumptions in these conjectures.

Findings

Standard simplex not most stable in Gaussian space for unequal measures.

Plurality not most stable in discrete space for unequal measures.

Results do not contradict original conjectures for equal measure partitions.

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 $k \geq 3$, for every value $\rho \neq 0$ of the noise and for every prescribed measures 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.