A key signature of MBL (many-body localization) is the slow rate at which information spreads. In this talk I will describe my recent results with Elgart showing that the infinite random XXZ spin-1/2 chain exhibits slow propagation of information (logarithmic light cone) in any arbitrary but fixed energy interval. The relevant parameter regime, which covers both weak interaction and strong disorder, is determined solely by the energy interval.

I will not assume that the audience is familiar with random spin chains. I will introduce the infinite random XXZ spin-1/2 chain, state the main result, and describe some important ingredients for the proof.