  To characterize the generic behavior of open quantum systems, we consider
random, purely dissipative Liouvillians with a notion of locality. We find that
the positivity of the map implies a sharp separation of the relaxation
timescales according to the locality of observables. Specifically, we analyze a
spin-1/2 system of size $\ell$ with up to $n$-body Lindblad operators, which
are $n$-local in the complexity-theory sense. Without locality ($n=\ell$), the
complex Liouvillian spectrum densely covers a "lemon"-shaped support, in
agreement with recent findings [Phys. Rev. Lett. 123, 140403;arXiv:1905.02155].
However, for local Liouvillians ($n<\ell$), we find that the spectrum is
composed of several dense clusters with random matrix spacing statistics, each
featuring a lemon-shaped support wherein all eigenvectors correspond to
$n$-body decay modes. This implies a hierarchy of relaxation timescales of
$n$-body observables, which we verify to be robust in the thermodynamic limit.
