The localized phase of pinning models with correlated Gaussian disorder

TL;DR

This paper extends key results of the pinning model with independent disorder to cases with correlated Gaussian disorder, using advanced probabilistic tools and assumptions on covariance structures.

Contribution

It generalizes the localized phase results of the pinning model to correlated Gaussian disorder, introducing new techniques for handling correlations.

Findings

Regularity of free energy is established under correlated disorder.

Central Limit Theorem for contact fraction is proven with correlated Gaussian disorder.

Results hold under summability of covariances and invertibility of the covariance operator.

Abstract

We show that most of the results proven in the localized regime of the pinning model with independent disorder (notably, $\mathcal{C}^\infty$ regularity of the free energy, size of the largest gap among pinned sites and Central Limit Theorem for the contact fraction) can be generalized to translation ergodic correlated disorder under the hypothesis that disorder is Gaussian. Most of the results, in particular $\mathcal{C}^\infty$ regularity and the Central Limit Theorem, are proven assuming only summability of the covariances. For some of the remaining main results we introduce the extra assumption that the covariance operator is invertible. The two key ingredients for the proof are the Birkhoff-sum approach introduced in~\cite{GZ25concentration} for independent disorder, but particularly adapted to handle correlated disorder, and decorrelation tools like the general and powerful Nelson's Gaussian hyper-contractivity and other tools that we develop and that are more specific to the one dimensional structure of the model we consider.