Low-Dimensional Life of Critical Anderson Electron

TL;DR

This paper demonstrates that the critical Anderson electron in three dimensions is confined to a fractal support of dimension approximately 8/3, implying its physics is effectively lower-dimensional and fully described by this support in infinite volume.

Contribution

It establishes that the critical Anderson electron's support is a fractal of dimension about 8/3, providing a new understanding of its spatial confinement and effective description.

Findings

Critical Anderson electron is confined to a fractal support of dimension ~8/3.

The physics of the critical electron is fully described by its lower-dimensional support.

Effective support dimension implies an exact spatial description in infinite volume.

Abstract

We show that critical Anderson electron in 3 dimensions is present in its spatial effective support, which was recently determined to be a region of fractal dimension $\approx \! 8/3$, with probability 1 in infinite volume. Hence, its physics is fully confined to space of this lower dimension. Stated differently, effective description of space occupied by critical Anderson electron becomes a full description in infinite volume. We then show that it is a general feature of the effective counting dimension underlying these concepts, that its subnominal value implies an exact description by effective support.