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

Abstract. We study the random pinning model, in the case of a Gaussian envi-ronment 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 νa is the same as the homogeneous one νpur, provided that correlations are decaying fast enough (a> 2). If correlations are summable (a> 1), we also show that the disordered phase tran-sition is at least of order 2, showing disorder relevance if νpur < 2. If correlations are not summable (a < 1), we show that the phase transition disappears. 1