Free-word association has been used as a vehicle to understand the organization of human thoughts. The original studies relied mainly on qualitative assertions, yielding the widely intuitive notion that trajectories of word associations are structured, yet considerably more random than organized linguistic text. Here we set to determine a precise characterization of this space, generating a large number of word association trajectories in a web implemented game. We embedded the trajectories in the graph of word co-occurrences from a linguistic corpus. To constrain possible transport models we measured the memory loss and the cycling probability. These two measures could not be reconciled by a bounded diffusive model since the cycling probability was very high (16 % of order-2 cycles) implying a majority of short-range associations whereas the memory loss was very rapid (converging to the asymptotic value in ∼ 7 steps) which, in turn, forced a high fraction of long-range associations. We show that memory loss and cycling probabilities of free word association trajectories can be simultaneously accounted by a model in which transitions are determined by a scale invariant probability distribution.