Property (T) in random quotients of hyperbolic groups at densities above 1/3

We study random quotients of a fixed non-elementary hyperbolic group in the Gromov density model. Let $G=⟨S\mathrm{\mid }T⟩$ be a finite presentation of a non-elementary hyperbolic group, and let $An{n}_{l,\omega }(G)$ be the set of elements of norm between $l-\omega (l)$ and $l$ in $G$. A random quotient at density $d$ and length $\omega$-near $l$ is defined by killing a uniformly randomly chosen set of $\mathrm{\mid }{S}_l(G){\mathrm{\mid }}^d$ words in $An{n}_{l,\omega (l)}(G)$, where $\omega (l)=o_l(l)$. We prove that for any d>1/3, such a quotient has Property (T) with probability tending to $1$ as $l$ tends to infinity. This result answers a question of Gromov–Ollivier and strengthens a theorem of .{Z}uk (c.f Kotowski–Kotowski).