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

We consider Hermitian random band matrices H in $d⩾1$ dimensions. The matrix elements $H_{xy},$ indexed by $x,y\in \Lambda \subset Z^d,$ 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.