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.
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