  The work of this thesis concerns the problem of linear low energy excitations
of vector spin glass models. An analytical and numerical study is carried out,
considering a fully connected random-field Heisenberg model at zero
temperature, a fully-connected vector p-spin glass model and a sparse
random-field Heisenberg model. We test these models against the low temperature
behavior of finite dimensional glassy systems, in particular we show that they
posses phases where the density of states is gapless with quasi-localised
modes. In the case of the sparse model, we show that the density of states
follows a quartic law at low frequency, consistently with several recent
measures of this quantity that can be found in the literature of computer
glasses. In all the three models, the spin glass transition is characterised in
terms of the behavior of the softest excitations. We found that in the fully
connected models the zero temperature spin glass transition in a field is a
delocalisation transition of the softest modes. In the sparse case, a weaker
form of delocalisation appears at the transition. These results broaden our
understanding of the zero temperature critical point, by showing how spin glass
ordering affects the way the system responds to small magnetic perturbations.
