We prove that random groups in Gromov density model at any density $d$ are with overwhelming probability either non-left-orderable or trivial. It implies the lack of left-orderability for $d<\frac{1}{2}$.
Random groups and Property (T): Żuk's theorem revisited
Random Groups are not $n$-Cubulated
Property (T) in random quotients of hyperbolic groups at densities above 1/3
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