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

Source 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 $α<\frac12$.