Superadditive correlation.

The fact that correlation does not imply causation is well known. Correlation between variables at two sites does not imply that the two sites directly interact, because, e.g., correlation between distant sites may be induced by chaining of correlation between a set of intervening, directly interacting sites. Such "noncausal correlation" is well understood in statistical physics: an example is long-range order in spin systems, where spins which have only short-range direct interactions, e.g., the Ising model, display correlation at a distance. It is less well recognized that such long-range "noncausal" correlations can in fact be stronger than the magnitude of any causal correlation induced by direct interactions. We call this phenomenon superadditive correlation (SAC). We demonstrate this counterintuitive phenomenon by explicit examples in (i) a model spin system and (ii) a model continuous variable system, where both models are such that two variables have multiple intervening pathways of indirect interaction. We apply the technique known as decimation to explain SAC as an additive, constructive interference phenomenon between the multiple pathways of indirect interaction. We also explain the effect using a definition of the collective mode describing the intervening spin variables. Finally, we show that the SAC effect is mirrored in information theory, and is true for mutual information measures in addition to correlation measures. Generic complex systems typically exhibit multiple pathways of indirect interaction, making SAC a potentially widespread phenomenon. This affects, e.g., attempts to deduce interactions by examination of correlations, as well as, e.g., hierarchical approximation methods for multivariate probability distributions, which introduce parameters based on successive orders of correlation.