Disorder strongly affects quantum dynamics by disrupting the coherent ballistic transport of clean systems. The prototype for this phenomenon is the Anderson model, governed by the Hamiltonian $H = -\Delta + V$ with $V$ a suitable random potential, which exhibits complete localization at strong disorder in any dimension. However, adding interactions complicates the picture: Many-body localization (MBL) is widely expected to persist in 1D and to be unstable over long time scales in higher dimensions. In this talk, I discuss small-velocity Lieb-Robinson bounds and other related bounds, which provide a way to quantify the disorder-induced slowing of dynamics in any dimension. I present two recent results proving the suppression of many-body transport: first, in the context of strongly disordered quantum spin systems, and second, for the bosonic Anderson model with mean-field interactions. I conclude with a complementary result, establishing delocalization for weakly disordered many-body models on sufficiently short, perturbative time scales.
Based on joint work with McDonough & Lucas, Rademacher & Zhang, and Toniolo.