Scaling limit for the pinning model in correlated Gaussian environment beyond the $L^2$-regime

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 $\alpha >0$. The covariance of the Gaussian environment $\{{\omega }_n{\}}_{n\in \mathbb{N}}$ is given by ${\text{Cov}}_{\mathbb{P}}({\omega }_n,{\omega }_m)\sim |n-m{|}^{2H-2}$ with $H\in (0,1)$. Assuming $\alpha \in (0,\frac{1}{2}]$, $H\in (\frac{1}{2},1)$ and $\alpha +2H>2$, 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 $\alpha <\frac{1}{2}$.

20 pages