Localisation and universal fluctuations in ultraslow diffusion processes

We study ultraslow diffusion processes with logarithmic mean squared displacement (MSD) 〈x² (t)〈 ≃ log^γt. Comparison of annealed (renewal) continuous time random walks (CTRWs) with logarithmic waiting time distribution ψ (τ) ≃ 1/(τ log^1+γ τ) and Sinai diffusion in quenched random landscapes reveals striking similarities, despite the great differences in their physical nature. In particular, they exhibit a weakly non-ergodic disparity of the time-averaged and ensemble-averaged MSDs. Remarkably, for the CTRW we observe that the fluctuations of time averages become universal, with an exponential suppression of mobile trajectories. We discuss the fundamental connection between the Golosov localization effect and non-ergodicity in the sense of the disparity between ensemble-averaged MSD and time-averaged MSD.