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

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{C