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The localized phase of pinning models with correlated Gaussian disorder

Giambattista Giacomin, Alexandre Legrand, Marco Zamparo

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.

The localized phase of pinning models with correlated Gaussian disorder

Abstract

We show that most of the results proven in the localized regime of the pinning model with independent disorder (notably, 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 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.

Paper Structure

This paper contains 18 sections, 22 theorems, 147 equations, 1 figure.

Table of Contents

  1. Localization in the pinning model
  2. The pinning model
  3. Main results
  4. Discussion of the results
  5. The alternative free energy
  6. Decay of correlations and largest jump in the localized phase
  7. Decorrelation lemma
  8. Polymer correlations in the localized phase
  9. Largest excursion, regularity and quenched CLT
  10. The Gaussian hyper-contractivity
  11. Largest jump
  12. Smoothness and CLT
  13. Variance of the random centering term
  14. Correlations under the environment measure
  15. Convergence of the variance
  16. ...and 3 more sections

Key Result

Theorem 1.3

The free energy $f$ is strictly convex and ${\mathcal{C}} ^\infty$ on $(h_c,+\infty)$, and for all $h>h_c$ and $r\in\mathbb{N}_0$, one has ${\mathbb P}$-a.s., Moreover $f$ is of class Gevrey-$3$ on $(h_c,+\infty)$, i.e. for every closed subset $H\subset (h_c,+\infty)$, there exists $c>0$ such that for all $r\in\mathbb{N}$, $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1:

Theorems & Definitions (49)

  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Remark 1.9
  • Lemma 2.1
  • proof
  • ...and 39 more