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It is demonstrated that a perturbation theory for the Coulomb Gap can be developed, which is identical to the TAP approach to spin glasses. Thus chapter 2 presents the standard theoretical method used to study the Coulomb Gap, together with the analogous result in spin-glass theory. Chapter 3 begins the exposition of new work contained in the thesis. The example of the classical dipole fluid is used to demonstrate how classical spin systems with long-ranged interactions can be expanded as a series of Feynman diagrams. A similar method, taking account of the disorder in the Coulomb Gap problem, is used to derive the strong-disorder Coulomb Gap density of states as a leading set of Feynman diagrams. In the next chapter, it is shown that the assumption of single-parameter scaling enables the use of the Renormalization Group to derive such non-perturbative effects as the crossover to weak disorder, and edge effects within the Coulomb Gap fluid. In chapter 5, it is shown that the analogy with spin glasses enables the use of the replica trick to obtain an effective field theory for the Coulomb Gap, which resembles the Sine-Gordon model in statistical physics. This resemblance allows a systematic derivation of the RG scaling function to any accuracy and a study of the renormalizability of the effective field theory, leading in principle to a test of the single-parameter scaling hypothesis. The second part of the thesis considers the optical phonons in a disordered ionic crystal, and in chapter 6 it is shown that the use of projection operators together with the standard techniques of perturbation theory enables a derivation of the phonon density of states. The result is that the longitudinal and transverse optic phonon modes are broadened by disorder into semicircular bands, and is identical to the result for the disorder-broadened Landau levels in a metal. Inspired by this similarity, methods of perturbation theory and supersymmetry have been applied to deriving the transport properties of the phonons. As described in section 7.2, the lack of a small parameter makes this impossible for the disordered ionic crystal, but the simpler problem of a uniaxial ionic crystal would be suitable for the use of these techniques.

The traditionally distinct fields of the structure of amorphous materials and time-frequency analysis are reviewed before time-frequency analysis is applied to the study of the diffraction data of amorphous materials for the first time. Several, general-purpose development in the field of wavelet-based time-frequency analysis, including new wavelets, are presented which were found to be necessary for the application of these methods to the study of diffraction data. A new representation of the diffraction data of amorphous materials is formulated in terms of the new method of time-frequency analysis. Additional new methods for the structural analysis of amorphous materials are also presented. These methods of analysis are applied to both experimentally-measured and computationally-simulated diffraction data for four different types of amorphous structure.  A detailed discussion is presented which examines various features found in the results of these analyses. In particular, the new representation of diffraction data is shown to be able to decompose the data over the real-reciprocal space plane. This is used to elucidate the existence of extended-range order in the amorphous structures, which is found to reproduce a simple functional form accurately in all cases. Additionally, different type of amorphous structure are shown to exhibit characteristic numbers of superimposed sets of these extended-range oscillations, which form the observed damped-density fluctuations. The relevance of existing mean-field theories is examined as a possible explanation for the functional form which we have observed empirically. Finally, conclusions are drawn based upon these results and suggestions are made for further work.

In this thesis measurements are presented of the frictional drag between two closely spaced but electrically isolated two dimensional electron gases, where the momentum transfer from a current in one layer is measured as a voltage in the adjacent layer. The drag measurement has been applied to this bilayer system under a number of different conditions. Firstly, at zero magnetic field and at temperatures comparable to the Fermi temperature, a new interlayer coupling mechanism is measured. The experimental traces are compared to a recent theory of the plasmon enhancement of the interlayer Coulomb coupling, which is shown to account for the observed behaviour. The drag is used to probe the bilayer plasmon modes, and it is shown that calculations in the random phase approximation provide an over estimate of the plasmon energies, which are reduced by many-body correlations. The fit to experiment is improved for calculations where the intra-layer exchange interaction is included in the Hubbard approximation. Next, the drag is measured in the presence of a magnetic field at temperatures sufficient to smear the Landau level structure. It is shown that the discrete energy spectrum of the electronic states leads to an enhancement of low energy scattering, which favours single particle Coulomb scattering as the dominant interlayer coupling. When the temperature is lowered into the QH regime, it is shown that the drag is more sensitive to the spin-splitting of the Landau levels than single layer transport measurements. It is also shown that the localised states in the Landau level tails, should be included in the theoretical models. A novel electron-hole symmetry applies to the interlayer scattering in this regime, and a scattering mechanism is suggested involving interedge scattering within each layer via localised states.

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