Role of dimensionality in the Anderson transition at strong disorder

Determine how spatial dimensionality influences the Anderson transition in the strong-disorder regime of Anderson localization, specifically characterizing the dependence of transition behavior and fluctuation properties on dimension and clarifying the organizing principles governing this regime.

Background

This paper establishes extensive numerical evidence that fluctuations in two-dimensional Anderson localization fall within the (1+1)-dimensional Kardar–Parisi–Zhang (KPZ) universality class, both for localized eigenstates and for long-time evolved wave packets. It further reveals glassy features analogous to directed polymer physics, such as dominant paths exhibiting pinning and avalanche behavior.

While the study provides a unified KPZ framework for 2D, the authors highlight that the situation in higher dimensions is richer and less understood. They point to the inherently more complex (d+1)-dimensional directed polymer problem and identify the need to clarify how dimensionality affects the Anderson transition under strong disorder, indicating this as an important open question currently under active debate.

References

This analogy with KPZ and directed polymer physics could help clarifying the role of dimensionality in the Anderson transition at strong disorder, an important open question actively debated .

— Kardar-Parisi-Zhang and glassy properties in 2D Anderson localization: eigenstates and wave packets  (2512.12085 - Izem et al., 12 Dec 2025) in Conclusions (Section 6), final paragraph

Whether these findings generalize beyond one dimension, however, remains unsettled, particularly for reciprocal non-Hermiticity realized through purely imaginary on-site disorder combined with real hopping.

— Enhanced Multifractality Induced by Non-Hermitian Disorder in Quantum Percolation  (2609.04625 - Oliveira et al., 4 Sep 2026) in Introduction

However, due to the presence of a large number sites at the boundary of HLs, which makes their effective dimensionality infinite, an essential singularity in TDOS of the form \rho_t(0) \sim \exp[-\lambda/\sqrt{W_{c,2}-W}], where \lambda is a constant, cannot be ruled out.

— Hyperbolic lattices with mass disorder: Phases and phase transitions  (2610.02192 - Sen et al., 1 Oct 2026) in Scaling section, paragraph beginning “Near the semimetal-to-metal QPT in DL”