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 maximise 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 three candidate elections, for all correlation parameters $\rho $ satisfying $00$ 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 $m=2$).

Keywords

• 60E15
• 49Q05
• 53A10