  In this work we investigate the generic properties of a stochastic linear
model in the regime of high-dimensionality. We consider in particular the
Vector AutoRegressive model (VAR) and the multivariate Hawkes process. We
analyze both deterministic and random versions of these models, showing the
existence of a stable and an unstable phase. We find that along the transition
region separating the two regimes, the correlations of the process decay
slowly, and we characterize the conditions under which these slow correlations
are expected to become power-laws. We check our findings with numerical
simulations showing remarkable agreement with our predictions. We finally argue
that real systems with a strong degree of self-interaction are naturally
characterized by this type of slow relaxation of the correlations.
