  We provide a full and rigorous proof of a theorem attributed to \.Zuk,
stating that random groups in the Gromov density model for d > 1/3 have
property (T) with high probability. The original paper had numerous gaps, in
particular, crucial steps involving passing between different models of random
groups were not described. We fix the gaps using combinatorial arguments and a
recent result concerning perfect matchings in random hypergraphs. We also
provide an alternative proof, avoiding combinatorial difficulties and relying
solely on spectral properties of random graphs in G(n, p) model.
