Correlation bound for distant parts of factor of IID processes

We study factor of i.i.d. processes on the $d$-regular tree for $d \geq 3$. We show that if such a process is restricted to two distant connected subgraphs of the tree, then the two parts are basically uncorrelated. More precisely, any functions of the two parts have correlation at most $k(d-1) / (\sqrt{d-1})^k$, where $k$ denotes the distance of the subgraphs. This result can be considered as a quantitative version of the fact that factor of i.i.d. processes have trivial 1-ended tails.