The disorder-relevance conjecture for correlated pinning models

Let $\tau$ and $\hat\tau$ be renewal sets with tail exponents

$$and$$

, respectively, and let $h_c(\beta)$ and $h_c^a(\beta)$ denote the quenched and annealed critical points of the pinning model with coupling strength $\beta$. Write $a\wedge b=\min\{a,b\}$. Disorder-relevance conjecture. Disorder is relevant for every $\beta>0$ in the sense of a critical-point shift if

$$\alpha>1-\frac{1}{\hat\alpha\wedge2},$$

and, in that case,

$$\limsup_{\beta\to0}\frac{\log\bigl(h_c(\beta)-h_c^a(\beta)\bigr)}{\log\beta}=\frac{(\alpha\wedge1)(\hat\alpha\wedge2)}{1-(\hat\alpha\wedge2)(1-(\alpha\wedge1))}.$$

This extends the predicted disorder-relevance criterion for pinning models to correlated renewal disorder; the exponent reflects the stable-law scaling of the disorder partial sums when $\hat\alpha\in(1,2)$. The conjecture is presented as a problem for future work, while the corresponding prediction for an i.i.d. $\gamma$-stable environment was proved by Lacoin and Sohier.