Probability distribution of the conductance at the mobility edge

TL;DR

This paper analyzes the probability distribution of conductance at the critical point of the metal-insulator transition in 3D and 4D systems, demonstrating dimension dependence and behavior at extreme conductance values.

Contribution

It provides a detailed analysis of the conductance distribution at the mobility edge, including proofs of dimension dependence and limiting behaviors.

Findings

Distribution P(g) depends on system dimension

P(g) approaches zero as g approaches zero

Form of P(g) is characterized at the critical point

Abstract

Distribution of the conductance P(g) at the critical point of the metal-insulator transition is presented for three and four dimensional orthogonal systems. The form of the distribution is discussed. Dimension dependence of P(g) is proven. The limiting cases $g\to\infty$ and $g\to 0$ are discussed in detail and relation $P(g)\to 0$ in the limit $g\to 0$ is proven.