We analyze the thermalization properties and the validity of the Eigenstate
Thermalization Hypothesis in a generic class of quantum Hamiltonians where the
quench parameter explicitly breaks a Z_2 symmetry. Natural realizations of such
systems are given by random matrices expressed in a block form where the terms
responsible for the quench dynamics are the off-diagonal blocks. Our analysis
examines both dense and sparse random matrix realizations of the Hamiltonians
and the observables. Sparse random matrices may be associated with local
quantum Hamiltonians and they show a different spread of the observables on the
energy eigenstates with respect to the dense ones. In particular, the numerical
data seems to support the existence of rare states, i.e. states where the
observables take expectation values which are different compared to the typical
ones sampled by the micro-canonical distribution. In the case of sparse random
matrices we also extract the finite size behavior of two different time scales
associated with the thermalization process.Comment: 30 pages, 44 figure