  Motivated by the constrained many-body dynamics, the stability of the
localization-delocalization properties to the inclusion of the soft constraints
is addressed in random matrix models. These constraints are modeled by
correlations in long-ranged hopping with Pearson correlation coefficient
different from zero or unity. Counterintuitive robustness of delocalized
phases, both ergodic and (multi)fractal, in these models is numerically
observed and confirmed by the analytical calculations. First, matrix inversion
trick is used to uncover the origin of such robustness. Next, to characterize
delocalized phases a method of eigenstate calculation, sensitive to
correlations in long-ranged hopping terms, is developed for random matrix
models and approved by numerical calculations and previous analytical ansatz.
The effect of the robustness of states in the bulk of the spectrum the
inclusion of to soft constraints is generally discussed for single-particle and
many-body systems.
