  In this work, we consider a model of a subsystem interacting with a reservoir
and study dynamics of entanglement assuming that the overall time-evolution is
governed by non-integrable Hamiltonians. We also compare to an ensemble of
Integrable Hamiltonians. To do this, we make use of unitary invariant ensembles
of random matrices with either Wigner-Dyson or Poissonian distributions of
energy. Using the theory of Weingarten functions, we derive universal average
time evolution of the reduced density matrix and the purity and compare these
results with several physical Hamiltonians: randomized versions of the
transverse field Ising and XXZ models, Spin Glass and, Central Spin and SYK
model. The theory excels at describing the latter two. Along the way, we find
general expressions for exponential $n$-point correlation functions in the gas
of GUE eigenvalues.
