This paper presents a simple model for such processes as spin diffusion or conduction in the "impurity band. " These processes involve transport in a lattice which is in some sense random, and in them diffusion is expected to take place via quantum jumps between localized sites. In this simple model the essential randomness is introduced by requiring the energy to vary randomly from site to site. It is shown that at low enough densities no diffusion at all can take place, and the criteria for transport to occur are given.

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Spin waves are studied in the t-J model for low dopant concentrations on the basis of a Green sfunction formalism in a slave-fermion Schwinger boson representation. The self-consistent Born approximation is used to calculate the Green s function for holes. Both the coupling of spin waves to electronhole pair excitations and the scattering of spin waves by holes are taken into account in calculating the Green s function for spin waves. The spin-wave velocity is evaluated for various values of J/t. For small values of J/t, it is found to be strongly renormalized due to the creation of electron-hole pairs.

A new basis has been found for the theory of localization of electrons in disordered systems. The method is based on a selfconsistent solution of the equation for the self energy in second order perturbation theory, whose solution may be purely real almost everywhere (localized states) or complex everywhere (nonlocalized states). The equations used are exact for a Bethe lattice. The selfconsistency condition gives a nonlinear integral equation in two variables for the probability distribution of the real and imaginary parts of the self energy. A simple approximation for the stability limit of localized states gives Anderson's 'upper limit approximation'. Exact solution of the stability problem in a special case gives results very close to Anderson's best estimate.