Correlation decay up to the sampling threshold in the local lemma regime

We study the decay of correlation between locally constrained independent random variables in the local lemma regimes. the distribution defined by constraint satisfaction problems (CSPs) in the local lemma regime. For atomically constrained independent random variables of sufficiently large domains, we show that a decay of correlation property holds up to the local lemma condition $pD^{2+o(1)}\lesssim 1$, asymptotically matching the sampling threshold for constraint satisfaction solutions [BGG+19,GGW22]. This provides evidence for the conjectured $pD^2\lesssim 1$ threshold for the "sampling Lov\'{a}sz local lemma". We use a recursively-constructed coupling to bound the correlation decay. Our approach completely dispenses with the "freezing" paradigm originated from Beck [Bec91], which was commonly used to deal with the non-self-reducibility of the local lemma regimes, and hence can bypass the current technical barriers due to the use of $\{2,3\}$-trees.