  We show that the nonlinear evolution of the cosmic gravitational clustering
is approximately spatial local in the $x$-$k$ (position-scale) phase space if
the initial perturbations are Gaussian. That is, if viewing the mass field with
modes in the phase space, the nonlinear evolution will cause strong coupling
among modes with different scale $k$, but at the same spatial area $x$, while
the modes at different area $x$ remain uncorrelated, or very weakly correlated.
We first study the quasi-local clustering behavior with the halo model, and
demonstrate that the quasi-local evolution in the phase space is essentially
due to the self-similar and hierarchical features of the cosmic gravitational
clustering. The scaling of mass density profile of halos insures that the
coupling between $(x-k)$ modes at different physical positions is substantially
suppressed. Using high resolution N-body simulation samples in the LCDM model,
we justify the quasi-locality with the correlation function between the DWT
(discrete wavelet transform) variables of the cosmic mass field. Although the
mass field underwent a highly non-linear evolution, and the DWT variables
display significantly non-Gaussian features, there are almost no correlations
among the DWT variables at different spatial positions. Possible applications
of the quasi-locality have been discussed.
