Standard Simplices and Pluralities are Not the Most Noise Stable

Steven Heilman, Elchanan Mossel, Joe Neeman

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 show a fundamental difference between noise stability of partitions into $k \geq 3$ parts and noise stability of partitions into $2$ parts. Moreover, our results further show a difference between noise stability of partitions into $k \geq 3$ parts and Gaussian surface area of partitions into $k \geq 3$ parts. Given our results it is natural to ask for (conjectured) partitions achieving the optimum noise stability.