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Anderson localization is the confinement of electron wave functions, or other wave modes, by interference between repeated wave scattering paths in static disorder. A localized eigenstate occupies a limited region even in a much larger sample. Disorder can therefore suppress diffusion without absorbing the wave. In 1958, Philip Warren Anderson studied random site energies and intersite coupling, finding a regime without transport. His paper develops a theoretical model; it does not report an Anderson laboratory measurement.^[ref]

The distinction from ordinary multiple scattering is the retention of wave phases: alternative paths contribute probability amplitudes before their squared modulus is taken. Adding path probabilities alone loses the cross terms responsible for interference. This makes Anderson localization useful as a warning about transport models: repeated deflections need not produce unlimited diffusion. It also connects electron transport with wave propagation in a random medium, although the detailed equations and experimental diagnostics differ between electrons, light and acoustic waves.

Philip Warren Anderson's 1977 lecture distinguishes localized magnetic moments from disorder-localized states. It credits Nevill Francis Mott with recognizing the significance of the mobility edge. A Mott insulator instead requires electron interactions and can exist without disorder.^[ref]

John Hasbrouck Van Vleck developed quantum mechanics explanations of magnetic susceptibility, including its temperature dependence; this is a distinct contribution from Anderson localization.^[ref]

In wave propagation in a random medium, the coherent field is the expectation of the complex wave amplitude, $m(\mathbf{r})=\mathbb{E}[E(\mathbf{r})]$. The remaining field $E-m$ has zero expectation. This is the coherent and diffuse wave fields decomposition. Statistical homogeneity of the medium does not make $m$ spatially constant when the incident field is localized: translation symmetry applies to the response together with its incident data. The coherent component can attenuate while the total ensemble-averaged wave intensity remains substantial.

Multiple scattering can enlarge the angular diversity reaching a receiver array. Reversing the paths then makes the array act like a larger aperture, improving the focus in appropriate regimes of wave propagation in a random medium.

A longitudinal Brownian field ${\mathcal{B}}_x(z)$ is a centered Gaussian random field withwhere $B$ is a valid transverse covariance kernel. For each fixed $z$ it is a scaled Brownian motion, and increments over disjoint longitudinal intervals are independent. Such a field drives a Markov model of wave propagation in a random medium. Its formal longitudinal derivative is Gaussian white noise in $x$, with transverse correlations described by $B$; it is not in general a Brownian sheet.

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.

Wave propagation in a random medium studies how fluctuations of properties such as refractive index, mass density or sound speed change an incident wave. Different paths acquire different wave phases, and spatial inhomogeneities generate wave scattering. One realization can therefore show a complicated transmitted pattern even under simple illumination.

The coherent field $⟨u⟩$ is the ensemble average of the field, while the mean wave intensity in a scalar normalization is $⟨|u{|}^2⟩$. They obeyThus decay of the coherent field need not mean loss of total wave energy: randomized wave phases can transfer weight to fluctuating contributions. Comparing a phase-sensitive average with an average of detector intensities tests this distinction. Weak fluctuations can permit perturbation methods, whereas repeated wave scattering with retained interference can lead to Anderson localization.