  In the context of $C^*$ dynamical systems, we consider a locally compact
group $G$ acting by $^*$-automorphisms on a C$^*$ algebra $\mathfrak{U}$ of
observables, and assume a state of $\mathfrak{U}$ that satisfies the clustering
property with respect to a net of group elements of $G$. That is, the two-point
connected correlation function vanishes in the limit on the net, when one
observable is translated under the group action. Then we show that all higher
order connected correlation functions (Ursell functions, or classical
cumulants) and all free correlation functions (free cumulants, from free
probability) vanish at the same rate in that limit. Additionally, we show that
mean clustering, also called ergodicity, extends to higher order correlations.
We then apply those results to equilibrium states of quantum spin lattice
models. Under certain assumptions on the range of the interaction, high
temperature Gibbs states are known to be exponentially clustering w.r.t. space
translations. Combined with the Lieb-Robinson bound, one obtains exponential
clustering for space-time translations outside the Lieb-Robinson light-cone.
Therefore, by our present results, all the higher order connected and free
correlation functions will vanish exponentially under space-time translations
outside the Lieb-Robinson light cone, in high temperature Gibbs states. Another
consequence is that their long-time averaging over a space-time ray vanishes
for almost every ray velocity.
