Three Candidate Plurality is Stablest for Small Correlations
share
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.
READ FULL TEXT