Three Candidate Plurality is Stablest for Small Correlations

11/11/2020
by   Steven Heilman, et al.
0

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.

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