Scaling limit for the pinning model in correlated Gaussian environment beyond the $L^2$-regime
Year: 2026
Type: preprint
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 $\{ω_n\}_{n\in\mathbb N}$ is given by $\text{Cov}_{\mathbb P}(ω_n,ω_m)\sim |n-m|^{2H-2}$ with $H\in(0,1)$. Assuming $α\in(0,\frac12]$, $H\in(\frac12,1)$ 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 $α<\frac12$.
Source: arXiv (Cornell University)
Authors: Song, Jian, Wang, Meng, Wei, Ran
Institutions: Shandong University, Tianjin University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G7526687548
OpenAlex ID: W7208793049