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On a pinning model in correlated Gaussian random environments

Zian Li, Jian Song, Ran Wei, Hang Zhang

TL;DR

This work analyzes a pinning model in a correlated Gaussian environment with covariance $\gamma(n) \sim |n|^{2H-2}$ and renewal exponent $\alpha$, identifying intermediate-disorder scaling limits. The authors construct Wick-ordered and non-Wick partition functions, prove Slkorohod (Wiener) chaos limits for the Wick-ordered case and Stratonovich chaos limits for the original model, under precise conditions on $H$ and $\alpha$ (notably $H\in(\tfrac12,1)$ and $\alpha>\tfrac12$, with stronger $\alpha+H>\tfrac32$ for Stratonovich). The limits are given by continuum chaos expansions with kernels $\phi_k$ and $\psi_{m,k}$ against Gaussian noise with covariance $|t-s|^{2H-2}$, aligning with the Weinrib–Halperin disorder-relevance predictions in the correlated setting. The paper also establishes uniform integrability and $L^1$-boundedness in stronger regimes, highlighting a new feature where full $L^2$-boundedness can fail in the disorder-relevant region due to environmental correlations, and discusses potential $L^1$ limits when $\alpha\le\tfrac12$. Overall, the results provide a rigorous link between discrete disordered pinning models and their continuum limits in correlated environments, enriching the understanding of disorder relevance beyond the classical Harris criterion.

Abstract

We consider a pinning model in correlated Gaussian random environments. For the model that is disorder relevant, we study its intermediate disorder regime and show that the rescaled partition functions converge to a non-trivial continuum limit in the Skorohod setting and in the Stratonovich setting, respectively. Our results partially confirm the prediction of Weinrib and Halperin for disorder relevance/irrelevance.

On a pinning model in correlated Gaussian random environments

TL;DR

This work analyzes a pinning model in a correlated Gaussian environment with covariance and renewal exponent , identifying intermediate-disorder scaling limits. The authors construct Wick-ordered and non-Wick partition functions, prove Slkorohod (Wiener) chaos limits for the Wick-ordered case and Stratonovich chaos limits for the original model, under precise conditions on and (notably and , with stronger for Stratonovich). The limits are given by continuum chaos expansions with kernels and against Gaussian noise with covariance , aligning with the Weinrib–Halperin disorder-relevance predictions in the correlated setting. The paper also establishes uniform integrability and -boundedness in stronger regimes, highlighting a new feature where full -boundedness can fail in the disorder-relevant region due to environmental correlations, and discusses potential limits when . Overall, the results provide a rigorous link between discrete disordered pinning models and their continuum limits in correlated environments, enriching the understanding of disorder relevance beyond the classical Harris criterion.

Abstract

We consider a pinning model in correlated Gaussian random environments. For the model that is disorder relevant, we study its intermediate disorder regime and show that the rescaled partition functions converge to a non-trivial continuum limit in the Skorohod setting and in the Stratonovich setting, respectively. Our results partially confirm the prediction of Weinrib and Halperin for disorder relevance/irrelevance.
Paper Structure (15 sections, 23 theorems, 151 equations, 2 figures)

This paper contains 15 sections, 23 theorems, 151 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The disordered pinning model
  3. Disorder relevance/irrelevance and the Weinrib-Halperin prediction
  4. Main results
  5. Preliminaries on Gaussian spaces and U-statistics
  6. Gaussian spaces
  7. U-statistics
  8. The Skorohod case
  9. Weak convergence
  10. Some discussion on uniform integrability
  11. The Stratonovich case
  12. --convergence of the continuum partition function.
  13. Weak convergence of the rescaled partition functions.
  14. --bound of the rescaled partition functions
  15. Some miscellaneous results

Key Result

Theorem 1.1

Let the renewal process $\tau$ and the disorder $\omega$ be given as in Section sec1.1. Let H and $\alpha$ be parameters satisfying Recalling the Wick ordered partition function $\tilde{Z}^{\omega,h}_{N,\beta}$ given in e:partition' and the scalings scale:beta, we have that where $\tilde{Z}_{\hat{\beta },\hat{h }}$ is given by the $L^2$-convergent series: where $\phi_k$ is given by e:phi, $\dot

Figures (2)

  • Figure 1: An example for $m=6$ and $j=2$ before modifications, the red segments represent the renewal kernels (with exponent $\alpha-1$) and the blue segments represent the kernels of the fractional Brownian motion (with exponent $2H-2$).
  • Figure 2: After modifications, the red segments represent the renewal kernels and the blue segments represent the kernels of the fractional Brownian motion.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 37 more