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

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.