Standard Simplices and Pluralities are Not the Most Noise Stable

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 discretenoise 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 inuence partition in discrete space for every number of parts k > 3, for every value ρ ≠ 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 concerning partitions into 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.In other words, the optimal partitions for noise stability are of a different nature than the ones considered for partitions into three parts in isoperimetric theory. In the latter case, the standard simplex is the partition of the plane into three sets of smallest Gaussian perimeter, where the sets are restricted to have Gaussian measures a1, a2, a3 > 0 respectively, with a1 + a2 + a3 = 1 and |ai --1/3|