Upper and Lower Bounds for The Quantum Dynamics of One-Dimensional Divergence-Type Random Jacobi Operators

We study quantum transport for the discrete one-dimensional random Jacobi operator of divergence-gradient type. For strictly positive and bounded random variables, we analyze the q-moments of the position operator and establish both upper and lower power-law bounds on their growth. Our approach relies on the asymptotic behavior of the integrated density of states and the Lyapunov exponent near the critical energy 0, previously obtained by Pastur and Figotin. A key ingredient in our analysis is the large deviation-type estimates explored via the phase formalism, which play a central role in deriving bounds on the growth of the transfer matrices.