Three Candidate Plurality is Stablest for Correlations at most 1/11

We prove the three candidate Plurality is Stablest Conjecture of Khot-Kindler-Mossel-O'Donnell from 2005 for correlations $\rho$ satisfying $-1/36<\rho<1/11$: the Plurality function is the most noise stable three candidate election method with small influences, when the corrupted votes have correlation $-1/36<\rho<1/11$ with the original votes. The previous best result of this type only achieved positive correlations at most $10^{-10^{10}}$. Our result follows by solving the three set Standard Simplex Conjecture of Isaksson-Mossel from 2011 for all correlations $-1/36<\rho<1/11$. The Gaussian Double Bubble Problem corresponds to the case $\rho\to1^{-}$, so in some sense, our result is a generalization of the Gaussian Double Bubble Problem. Our result is also notable since it is the first result for any $\rho<0$, which is the only relevant case for computational hardness of MAX-3-CUT. As an additional corollary, we conclude that three candidate Borda Count is stablest for all $-1/36<\rho<1/11$.Comment: 38 page