Characterize the disorder-irrelevant regime

Determine whether the parameter region $\{(\alpha,H):\alpha\in(0,1/2],\ \alpha+H<1\}$ is disorder-irrelevant and develop a criterion for disorder irrelevance when $H>1/2$.

Background

The Weinrib–Halperin prediction conjectures that the gray region defined by α∈(0,1/2]\alpha\in(0,1/2] and α+H<1\alpha+H<1 is disorder-irrelevant. The paper does not establish this prediction.

For H>1/2H>1/2, the usual criterion based on comparing quenched-free-energy critical exponents cannot be used because the quenched free energy has no critical exponent. Consequently, a new criterion is needed. The authors also note that the expected results for H<1/2H<1/2 have not yet been established in the relevant setting.

References

It is conjectured by the Weinrib--Halperin prediction that the gray regime in the phase diagram~\ref{Fig:phase-plane} \begin{equation*} \Big{(\alpha,H): \alpha\in\Big(0,\frac12\Big], \alpha+H<1\Big} \end{equation*} is disorder-irrelevant. The study of this regime remains largely open.

— 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 “Disorder-irrelevant regime”