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We study the nature of the vibrational modes in a two-dimensional harmonic lattice with long-range correlated random masses, with power-law spectral density S(k)∼1/kα. We obtain numerically the scale invariance of the fluctuations of the relative participation number and the local density of states. We find signatures of extended vibrational modes when α>αc and αc depends on the magnitude of disorder. In order to confirm this claim, we also study the time evolution of an initially localized perturbation of the lattice. We show that the second moment of the spatial distribution of the energy displays a ballistic regime when α>αc, in agreement with the occurrence of extended vibrational modes. [ABSTRACT FROM AUTHOR]