Using Gaussian integral transform techniques borrowed from functional-integral field theory and the replica trick we derive a version of the coherent-potential approximation (CPA) suited for describing ($i$) the diffusive (hopping) motion of classical particles in a random environment and ($ii$) the vibrational properties of materials with spatially fluctuating elastic coefficients in topologically disordered materials. The effective medium in the present version of the CPA is not a lattice but a homogeneous and isotropic medium, representing an amorphous material on a mesoscopic scale. The transition from a frequency-independent to a frequency-dependent diffusivity (conductivity) is shown to correspond to the boson peak in the vibrational model. The anomalous regimes above the crossover are governed by a complex, frequency-dependent self energy. The boson peak is shown to be stronger for non-Gaussian disorder than for Gaussian disorder. We demonstrate that the low-frequency non-analyticity of the off-lattice version of the CPA leads to the correct long-time tails of the velocity autocorrelation function in the hopping problem and to low-frequency Rayleigh scattering in the wave problem. Furthermore we show that the present version of the CPA is capable to treat the percolative aspects of hopping transport adequately.
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