Property (T) in density-type models of random groups

We study property (T) in the Gamma(n, k, d) model of random groups: as k tends to infinity this gives the Gromov density model, introduced in 1993. We provide bounds for property (T) in the k-angular model of random groups, i.e., the Gamma(n, k, d) model, where k is fixed and n tends to infinity. We also prove that for d > 1/3, a random group in the Gamma(n, k, d) model has property (T) with probability tending to 1 as k tends to infinity, strengthening the results of Zuk and Kotowski-Kotowski, who consider only groups in the Gamma(n, 3k, d) model.