The localized phase of pinning models with correlated Gaussian disorder

Abstract

We demonstrate that the results established for the localized regime of the pinning model with independent disorder -- notably the $\mathcal{C}^\infty$ regularity of the free energy, the scaling of the largest excursion between pinned sites, and the Central Limit Theorems for both the contact number and its mean -- can be generalized to translation-ergodic environments under the hypothesis that disorder is Gaussian. Our results are obtained assuming only summability of the charges' covariances. The two key ingredients for our proofs are the Birkhoff-sum approach introduced in Giacomin, Zamparo (2024) for independent disorder, but uniquely suited to handle correlated environments, and decorrelation tools such as the general and powerful Nelson's Gaussian hypercontractivity. We also rely on the Bernstein blocking method to prove a Central Limit Theorem under dependence. Additionally, we develop further techniques specifically tailored to the one-dimensional structure of the pinning model.