Experiments in Rydberg atoms have recently found unusually slow decay from a
small number of special initial states. We investigate the robustness of such
long-lived states (LLS) by studying an ensemble of locally constrained random
systems with tunable range μ. Upon varying μ, we find a transition
between a thermal and a weakly non-ergodic (supporting a finite number of LLS)
phases. Furthermore, we demonstrate that the LLS observed in the experiments
disappear upon the addition of small perturbations so that the transition
reported here is distinct from known ones. We then show that the LLS dynamics
explores only part of the accessible Hilbert space, thus corresponding to
localisation in Hilbert space.Comment: 5 pages, 3 figures + Supp. Ma