  Percolation theory dictates an intuitive picture depicting correlated regions
in complex systems as densely connected clusters. While this picture might be
adequate at small scales and apart from criticality, we show that highly
correlated sites in complex systems can be inherently disconnected. This
finding indicates a counter-intuitive organization of dynamical correlations,
where functional similarity decouples from physical connectivity. We illustrate
the phenomena on the example of the Disordered Contact Process (DCP) of
infection spreading in heterogeneous systems. We apply numerical simulations
and an asymptotically exact renormalization group technique (SDRG) in 1, 2 and
3 dimensional systems as well as in two-dimensional lattices with long-ranged
interactions. We conclude that the critical dynamics is well captured by mostly
one, highly correlated, but spatially disconnected cluster. Our findings
indicate that at criticality the relevant, simultaneously infected sites
typically do not directly interact with each other. Due to the similarity of
the SDRG equations, our results hold also for the critical behavior of the
disordered quantum Ising model, leading to quantum correlated, yet spatially
disconnected, magnetic domains.
