  Information-theoretic quantities such as Renyi entropies show a remarkable
universality in their late-time behaviour across a variety of chaotic quantum
many-body systems. Understanding how such common features emerge from very
different microscopic dynamics remains an important challenge. In this work, we
address this question in a class of Brownian models with random time-dependent
Hamiltonians and a variety of different microscopic couplings. In any such
model, the Lorentzian time-evolution of the $n$-th Renyi entropy can be mapped
to evolution by a Euclidean Hamiltonian on 2$n$ copies of the system. We
provide evidence that in systems with no symmetries, the low-energy excitations
of the Euclidean Hamiltonian are universally given by a gapped
quasiparticle-like band. The eigenstates in this band are plane waves of
locally dressed domain walls between ferromagnetic ground states associated
with two permutations in the symmetric group $S_n$. These excitations give rise
to the membrane picture of entanglement growth, with the membrane tension
determined by their dispersion relation. We establish this structure in a
variety of cases using analytical perturbative methods and numerical
variational techniques, and extract the associated dispersion relations and
membrane tensions for the second and third Renyi entropies. For the third Renyi
entropy, we argue that phase transitions in the membrane tension as a function
of velocity are needed to ensure that physical constraints on the membrane
tension are satisfied. Overall, this structure provides an understanding of
entanglement dynamics in terms of a universal set of gapped low-lying modes,
which may also apply to systems with time-independent Hamiltonians.
