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 $α&gt;0$. The covariance of the Gaussian environment ${ω<em>n}</em>{n\in\mathbb N}$ is given by $$\text{Cov}_{\mathbb P}(ω_n,ω_m)\sim |n-m|<sup>{2H-2}$$ with $H\in(0,1)$. Assuming $α\in(0,\frac12]$, $H\in(\frac12,1)$ and $α+2H&gt;2$, we show that the partition function of the disordered pinning model, under the appropriate scaling, converges in distribution to the $L<sup>1$-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 $L<sup>2$-integrable when $α&lt;\frac12$.