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PreprintarXiv licencearXiv

Scaling limit for the pinning model in correlated Gaussian environment beyond the L2-regime

  • Jian Song
  • Meng Wang
  • Ran Wei
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Abstract

In this paper, we study the scaling limit of the pinning model in correlated Gaussian environment. The tail probability of the underlying renewal process of the model has a polynomial decay with exponent α>0. The covariance of the Gaussian environment {ωn}n∈ℕ is given by Covℙ(ωn,ωm)∼ |n−m|2H−2 with H∈(0,1). Assuming α∈(0,1/2], H∈(1/2,1) and α+2H>2, we show that the partition function of the disordered pinning model, under the appropriate scaling, converges in distribution to the L1-solution of the fractional stochastic heat equation driven by Gaussian noise correlated in time and localized at the origin. In particular, it is known that the solution is not L2-integrable when α<1/2.

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Cite

BibTeXRIS
BibTeX
@misc{song2026scaling,
  title = {{Scaling limit for the pinning model in correlated Gaussian environment beyond the $L^2$-regime}},
  author = {Jian Song and Meng Wang and Ran Wei},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03607},
  eprint = {2609.03607},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03607},
}
RIS
TY  - UNPB
TI  - Scaling limit for the pinning model in correlated Gaussian environment beyond the L^2-regime
AU  - Jian Song
AU  - Meng Wang
AU  - Ran Wei
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03607
UR  - https://arxiv.org/abs/2609.03607
AB  - In this paper, we study the scaling limit of the pinning model in correlated Gaussian environment. The tail probability of the underlying renewal process of the model has a polynomial decay with exponent α>0. The covariance of the Gaussian environment {ω_n}_n∈ℕ is given by Cov_ℙ(ω_n,ω_m)∼ |n−m|^2H−2 with H∈(0,1). Assuming α∈(0,1/2], H∈(1/2,1) and α+2H>2, we show that the partition function of the disordered pinning model, under the appropriate scaling, converges in distribution to the L^1-solution of the fractional stochastic heat equation driven by Gaussian noise correlated in time and localized at the origin. In particular, it is known that the solution is not L^2-integrable when α<1/2.
ER  -