Comments on the influence of disorder for pinning model in correlated Gaussian environment

We study the random pinning model, in the case of a Gaussian environment presenting power-law decaying correlations, of exponent decay a > 0. A similar study was done in a hierachical version of the model Berger and Toninelli (2013), and we extend here the results to the non-hierarchical (and more natural) case. We comment on the annealed (i.e. averaged over disorder) model, which is far from being trivial, and we discuss the influence of disorder on the critical properties of the system. We show that the annealed critical exponent. a is the same as the homogeneous one nu(pur), provided that correlations are decaying fast enough (a > 2). If correlations are summable (a > 1), we also show that the disordered phase transition is at least of order 2, showing disorder relevance if nu(pur) < 2. If correlations are not summable (a < 1), we show that the phase transition disappears.