We investigate the weakly nonlinear evolution of cosmic gravitational clustering in phase space by looking at the Zeldovich solution in the discrete wavelet transform (DWT) representation. We show that if the initial perturbations are Gaussian, the relation between the evolved DWT mode and the initial perturbations in the weakly nonlinear regime is quasi-local. That is, the evolved density perturbations are mainly determined by the initial perturbations localized in the same spatial range. Furthermore, we show that the evolved mode is monotonically related to the initial perturbed mode. Thus, large (small) perturbed modes statistically correspond to the large (small) initial perturbed modes. We test this prediction by using quasi-stellar object Lyα absorption samples. The results show that the weakly nonlinear features for both the transmitted flux and the identified forest lines are quasi-localized. The locality and monotonic properties provide a solid basis for the DWT scale-by-scale Gaussianization reconstruction algorithm proposed by L.-L. Feng & L.-Z. Fang for data in the weakly nonlinear regime. With the Zeldovich solution, we also find that the major non-Gaussianities caused by the weakly nonlinear evolution are local scale-scale correlations. Therefore, to have a precise recovery of the initial Gaussian mass field, it is essential to remove the scale-scale correlations.
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