On Approximability of Satisfiable k-CSPs: IV

We prove a stability result for general 3-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if Σ,Γ and Φ are alphabets of constant size, and µ is a distribution over Σ×Γ×Φ satisfying: (1) the probability of each atom is at least Ω(1), (2) µ is pairwise connected, and (3) µ has no Abelian embeddings into (ℤ,+), then the following holds. Any triplets of 1-bounded functions f∶ Σn→ℂ, g∶ Γn→ℂ, h∶ Φn→ℂ satisfying