  We define auto- and cross-correlation functions capable to capture dynamical
characteristics induced by local phase space structures in a general dynamical
system. These correlation functions are calculated in the Standard Map for a
range of values of the non-linearity parameter $k$. Using a model of
non-interacting particles, each evolving according to the same Standard Map
dynamics and located initially at specific phase space regions, we show that
for $0.6 < k \leq 1.2$ long-range cross-correlations emerge. They occur as an
ensemble property of particle trajectories by an appropriate choice of the
phase space cells used in the statistical averaging. In this region of
$k$-values the single particle phase space is either dominated by local chaos
($k \leq k_c$ with $k_c \approx 0.97$) or it is characterized by the transition
from local to global chaos ($k_c < k \leq 1.2$). Introducing suitable symbolic
dynamics we demonstrate that the emergence of long-range cross-correlations can
be attributed to the existence of an effective intermittent dynamics in
specific regions of the phase space. Our findings further support the recently
found relation of intermittent dynamics with the occurrence of
cross-correlations (F.K. Diakonos, A.K. Karlis, and P. Schmelcher, Europhys.
Lett. {\bf 105}, 26004 (2014)) in simple one-dimensional intermittent maps,
suggesting its validity for two-dimensional Hamiltonian systems.
