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 $\alpha>0$. 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 $\alpha\in(0,\frac12]$, $H\in(\frac12,1)$ 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 $\alpha<\frac12$.