A random medium is a material whose spatially varying properties are described by random fields. For example,models a refractive index as a mean plus a fluctuation. A single specimen has one realized index profile; the statistical model describes an ensemble of possible profiles. Its covariance and correlation length specify how strongly nearby fluctuations are related.
Statistical homogeneity means that the joint statistics are invariant under translation, while statistical isotropy adds invariance under rotation. Neither assumption means that the index is constant in one specimen. A random medium is useful when measuring every microscopic inhomogeneity is impractical but statistical predictions are reproducible. Wave propagation in a random medium then distinguishes an ensemble-averaged coherent field from the mean wave intensity; the first alone does not determine the second.
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