Extend the disorder-relevant regime beyond the L1 regime established in the paper

Establish disorder relevance throughout the parameter region with $\alpha+H>1$, $\alpha+2H\leq2$, $\alpha\in(0,1/2]$, and $H\in(1/2,1)$, and determine whether this region is an $L^1$ regime analogous to the region with $\alpha+2H>2$.

Background

The main theorem proves convergence to an L1L^1 solution of the fractional stochastic heat equation when α+2H>2\alpha+2H>2 and α≤1/2\alpha\leq1/2. The Weinrib–Halperin prediction instead places the disorder-relevant threshold at α+H>1\alpha+H>1.

The intermediate region where α+H>1\alpha+H>1 but α+2H≤2\alpha+2H\leq2 is therefore not covered by the paper’s method. The authors explain that extending the result requires estimates that do not rely on finiteness of the self-energy, and they explicitly state that it is unknown whether this region has the same L1L^1 structure as the solved regime.

References

Hence, we believe that on the regime (the orange regime in the phase diagram \ref{Fig:phase-plane}) \begin{equation*} \Big{(\alpha,H):\alpha+H>1,\alpha+2H\leq2, \alpha\in\Big(0,\frac12\Big], H\in\Big(\frac12,1\Big)\Big}, \end{equation*} disorder is relevant, but it is still not clear whether it is an $L1$-regime as $\alpha+2H>2$.

— Scaling limit for the pinning model in correlated Gaussian environment beyond the $L^2$-regime  (2609.03607 - Song et al., 3 Sep 2026) in Section 1.4, Further discussion, subsection “The condition $\alpha+2H>2$”