# Localization in random fractal lattices

## TL;DR

This paper studies how eigenfunction localization in random fractal lattices depends on spectral and Hausdorff dimensions, revealing that spectral dimension critically influences localization while Hausdorff dimension has minimal effect.

## Contribution

It demonstrates that spectral dimension governs localization properties in random fractal lattices with fixed Hausdorff dimension, highlighting the role of connectivity over fractal geometry.

## Key findings

- Localization depends strongly on spectral dimension.
- Hausdorff dimension has little effect on localization.
- Superlocalization resonances and energy gaps occur at low spectral dimensions.

## Abstract

We investigate the issue of eigenfunction localization in random fractal lattices embedded in two dimensional Euclidean space. In the system of our interest, there is no diagonal disorder -- the disorder arises from random connectivity of non-uniformly distributed lattice sites only. By adding or removing links between lattice sites, we change the spectral dimension of a lattice but keep the fractional Hausdorff dimension fixed. From the analysis of energy level statistics obtained via direct diagonalization of finite systems, we observe that eigenfunction localization strongly depends on the spectral dimension. Conversely, we show that localization properties of the system do not change significantly while we alter the Hausdorff dimension. In addition, for low spectral dimensions, we observe superlocalization resonances and a formation of an energy gap around the center of the spectrum.