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

We consider Hermitian random band matrices H in \(d \geqslant 1 \) dimensions. The matrix elements \(H_{xy},\) indexed by \(x, y \in \varLambda \subset \mathbb {Z}^d,\) 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 \(|\varLambda | \) of the matrix.