Transfer matrices and conductivity in two- and three-dimensional systems. I. Formalism

The electronic states responsible for transport constitute a wildly fluctuating minority of the total number of states in a disordered system. These fluctuations are the essence of the problem and demand a rich and powerful formulation if they are to be described accurately. In this paper the author introduces a new formalism for treating transport in three-dimensional systems based on the transfer matrix. The method gains its power from the interplay of theory of the symmetric group, which enables irrelevant parts of the mathematics to be thrown out, leaving the essentials intact but simplified. Analogies with replica theory are drawn, and replica symmetry breaking discussed.