The localized phase of pinning models with correlated Gaussian disorder
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arXiv
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| Format: | Preprint |
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2026
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| author | Giacomin, Giambattista Legrand, Alexandre Zamparo, Marco |
| author_facet | Giacomin, Giambattista Legrand, Alexandre Zamparo, Marco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01137 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The localized phase of pinning models with correlated Gaussian disorder Giacomin, Giambattista Legrand, Alexandre Zamparo, Marco Probability 60K37, 82B44, 60K35, 60F05 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. |
| title | The localized phase of pinning models with correlated Gaussian disorder |
| topic | Probability 60K37, 82B44, 60K35, 60F05 |
| url | https://arxiv.org/abs/2604.01137 |