Dynamical Localization for Random Band Matrices up to $W\ll N^{1/4}$

TL;DR

This paper proves that certain Gaussian random band matrices exhibit dynamical Anderson localization at all energies when the band width is significantly smaller than the quarter power of the matrix size, using advanced probabilistic methods.

Contribution

It establishes dynamical localization for a broad class of random band matrices with band width up to $N^{1/4}$, extending previous results to larger matrices.

Findings

Dynamical localization holds for $W \

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Abstract

We prove that a large class of $N\times N$ Gaussian random band matrices with band width $W$ exhibits dynamical Anderson localization at all energies when $W \ll N^{1/4}$. The proof uses the fractional moment method and an adaptive Mermin--Wagner style shift.