Three Candidate Plurality is Stablest for Small Correlations

Abstract

Using the calculus of variations, we prove the following structure theorem for noise stable partitions: a partition of n-dimensional Euclidean space into m disjoint sets of fixed Gaussian volumes that maximize their noise stability must be (m − 1)-dimensional, if m − 1 ≤ n. In particular, the maximum noise stability of a partition of m sets in R n of fixed Gaussian volumes is constant for all n satisfying n ≥ m − 1. From this result, we obtain:(i) A proof of the Plurality is Stablest Conjecture for 3 candidate elections, for all correlation parameters ρ satisfying 0 < ρ < ρ 0 , where ρ 0 > 0 is a fixed constant (that does not depend on the dimension n), when each candidate has an equal chance of winning. (ii) A variational proof of Borell's Inequality (corresponding to the case m = 2). The structure theorem answers a question of De-Mossel-Neeman and of Ghazi-Kamath-Raghavendra. Item (i) is the first proof of any case of the Plurality is Stablest Conjecture of Khot-Kindler-Mossel-O'Donnell (2005) for fixed ρ, with the case ρ → 1 − being solved recently. Item (i) is also the first evidence for the optimality of the Frieze-Jerrum semidefinite program for solving MAX-3-CUT, assuming the Unique Games Conjecture. Without the assumption that each candidate has an equal chance of winning in (i), the Plurality is Stablest Conjecture is known to be false.

A Variational Proof of Robust Gaussian Noise Stability

“…The combination of our robust Borell inequality, Theorem 1.6 below, with previous works such as [BBJ17,Hei19,HT20] essentially shows that one single argument can prove nearly every known inequality for sets or partitions that maximize noise stability, with respect to Gaussian volume constraints. So, instead of having disparate arguments to prove these inequalities, one single calculus of variations argument has emerged, providing an aesthetically pleasing way to prove these optimal inequalities.…”

“…Moreover, the proof methods of [MN15b,Eld15] do not seem to generalize to inequalities for the noise stability of partitions of Euclidean space, as opposed to the calculus of variations arguments of e.g. [Hei19,HT20].…”

Unique Games hardness of Quantum Max-Cut, and a vector-valued Borell's inequality

“…The classical case, due to Borell [Bor85], is when k = 1; in this case f opt is just defined as f opt (x) = sgn(x 1 ), and the quantity E u∼ρv [f (u)f (v)] is known as the Gaussian noise stability of f . When the sphere S k−1 is replaced by the probablity simplex ∆ k−1 , the analogue of Theorem 1.2 is a well-known open problem-known as the "Peace Sign Conjecture" [IM12]-that was recently solved when ρ is sufficiently close to zero [HT20].…”

“…Arguments of this kind go back to McGonagle and Ross [MR15] in the setting of the Gaussian isoperimetric problem. They were developed in the vector-valued (but still isoperimetric) setting by Milman and Neeman [MN18], and then applied to noise stability by Heilman and Tarter [HT20].…”

Probabilistic Aspects of Voting, Intransitivity and Manipulation

“…On the other hand, in the balanced case, for q = 3, and for small positive values of ρ, the conjectured Gaussian Double-Bubble was very recently shown to be CHAPTER 6. VERY RECENT PROGRESS optimal [39].…”