Phase transitions for random states and a semi-circle law for the partial transpose

5 pages, 2 figures. This short note contains a high-level overview of two long and technical papers, arXiv:1011.0275 and arXiv:1106.2264. Version 2: unchanged results, editorial changes, added reference, close to the published articleInternational audienceFor a system of N identical particles in a random pure state, there is a threshold k_0 = k_0(N) ~ N/5 such that two subsystems of k particles each typically share entanglement if k > k_0, and typically do not share entanglement if k k_1. Since, for a given state of the entire system, the induced state of a subsystem is given by the partial trace, the above facts can be rephrased as properties of random induced states. An important step in the analysis depends on identifying the asymptotic spectral density of the partial transposes of such random induced states, a result which is interesting in its own right