7 - The Scaling Theory of Localization

Abstract

This chapter provides an overview of the scaling theory of localization. The scaling hypothesis is a powerful assumption from which many localization properties may be deduced. One way to check its validity is via numerical simulation, which has gained acceptance over the past several years as a means of verifying theoretical hypothesis and predicting the consequences of theories where analytical results are impossible or not yet available. As numerical simulation can deal only with finite samples, the variation of γ with a finite sample size L is a near-perfect case for numerical simulation. In one-dimensional, β (ln γ) may be written down explicitly, and numerical simulations in two-dimensional and three-dimensional can be carried out by a scheme, commonly known as the finite-size scaling approach, which is a generalization of the one-dimensional problem to systems of finite cross sections.