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 $m-1\leq n$. In particular, the maximum noise stability of a partition of $m$ sets in $\mathbb{R}^{n}$ of fixed Gaussian volumes is constant for all $n$ satisfying $n\geq m-1$. From this result, we obtain: (i) A proof of the Plurality is Stablest Conjecture for $3$ candidate elections, for all correlation parameters $\rho$ satisfying $0<\rho<\rho_{0}$, where $\rho_{0}>0$ 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 $\rho$, with the case $\rho\to1^{-}$ 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.