Convergence in High Probability of the Quantum Diffusion in a Random Band Matrix Model

TL;DR

This paper proves that quantum diffusion in high-dimensional random band matrices converges with high probability uniformly across matrix sizes, extending previous expectation-based results to a stronger probabilistic form.

Contribution

It advances the understanding of quantum diffusion by establishing high probability convergence in random band matrices, improving upon prior expectation-based results.

Findings

Quantum diffusion converges with high probability in random band matrices.

Convergence is uniform across matrix sizes.

Results extend previous expectation-based convergence to high probability.

Abstract

We consider Hermitian random band matrices $H$ in $d \geq 1 $ dimensions. The matrix elements $H_{xy},$ indexed by $x, y \in \Lambda \subset \mathbb{Z}^d,$ are independent, uniformly distributed random variable if $|x-y| $ is less than the band width $W,$ and zero otherwise. We update the previous results of the converge of quantum diffusion in a random band matrix model from convergence of the expectation to convergence in high probability. The result is uniformly in the size $|\Lambda| $ of the matrix.