Standard simplices and pluralities are not the most noise stable

Abstract

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 ≥ 3, for every value ρ = 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.

Three candidate plurality is stablest for small correlations

“…Conjecture 1.2 for all fixed parameters 0 < < 1 was entirely open until now. Unlike the case of the majority is stablest (Theorem 1.8), Conjecture 1.2 cannot hold when the candidates have unequal chances of winning the election [HMN16]. This realization is an obstruction to proving Conjecture 1.2.…”

“…This proof method can also prove Borell's original inequality [HT21,Hei21]. The result of [HT21] also circumvented a difficulty identified in [HMN16]. Although Theorem 1.2 holds when the average value of f is fixed to be some number −1 < a < 1, the three candidate analogue of the Majority is Stablest Theorem can only be true when the voting method takes all of its values with equal probability.…”

“…. , 1/ ) [HMN16]. In the remaining case that = 1/ for all 1 ≤ ≤ , it is assumed that = 0 in Conjecture 1.6.…”

“…For an even more general version of Theorem 1.8, see [MOO10,Theorem 4.4]. In particular, the assumption on E can be removed, though we know that this cannot be done for ≥ 3 [HMN16].…”

Untitled

“…This was explicitly conjectured in [20], in the special case that all of the parts in the partition have equal measure. However, a somewhat surprising recent result [14] showed that the standard simplex partition fails to maximize multi-part noise stability unless all of the parts have equal measure. On the other hand, there is also some support for the conjecture in the equal-measure case: Heilman [13] showed that the conjecture is true if the noise rate t is larger than some explicit function of the ambient dimension n.…”