Quantum transmission in disordered insulators: random matrix theory and transverse localization

We consider quantum interferences of classically allowed or forbidden electronic trajectories in disordered dielectrics. Without assuming a directed path approximation, we represent a strongly disordered elastic scatterer by its transmission matrix ${\bf t}$. We recall how the eigenvalue distribution of ${\bf t.t}^{\dagger}$ can be obtained from a certain ansatz leading to a Coulomb gas analogy at a temperature $\beta^{-1}$ which depends on the system symmetries. We recall the consequences of this random matrix theory for quasi--$1d$ insulators and we extend our study to microscopic three dimensional models in the presence of transverse localization. For cubes of size $L$, we find two regimes for the spectra of ${\bf t.t}^{\dagger}$ as a function of the localization length $\xi$. For $L / \xi \approx 1 - 5$, the eigenvalue spacing distribution remains close to the Wigner surmise (eigenvalue repulsion). The usual orthogonal--unitary cross--over is observed for {\it large} magnetic field change $\Delta B \approx \Phi_0 /\xi^2$ where $\Phi_0$ denotes the flux quantum. This field reduces the conductance fluctuations and the average log--conductance (increase of $\xi$) and induces on a given sample large magneto--conductance fluctuations of typical magnitude similar to the sample to sample fluctuations (ergodic behaviour). When $\xi$ is of the order of the