  The Glauber dynamics of various models (REM-like trap models, Brownian
motion, BM model, Ising chain and SK model) is analyzed in relation with the
existence of ageing. From a finite size Glauber matrix, we calculate a time
$\tau_w(N)$ after which the system has relaxed to the equilibrium state. The
case of metastability is also discussed. If the only non zero overlaps between
pure states are only self-overlaps (REM-like trap models, BM model), the
existence or absence of ageing depends only on the behavior of the density of
eigenvalues for small eigenvalues. We have carried out a detailed numerical and
analytical analysis of the density of eigenvalues of the REM-like trap models.
In this case, we show that the behavior of the density of eigenvalues for
typical trap realizations is related to the spectral dimension of the
equivalent random walk model.
