Marginal relevance for the $\gamma$-stable pinning model

We investigate disorder relevance for the pinning of a renewal when the law of the random environment is in the domain of attraction of a stable law with parameter $\gamma \in (1,2)$. Assuming that the renewal jumps have power-law decay, we determine under which condition the critical point of the system modified by the introduction of a small quantity of disorder. In an earlier study of the problem, we have shown that the answer depends on the value of the tail exponent $\alpha$ associated to the distribution of renewal jumps: when $\alpha>1-\gamma^{-1}$ a small amount of disorder shifts the critical point whereas it does not when $\alpha<1-\gamma^{-1}$. The present paper is focused on the boundary case $\alpha=1-\gamma^{-1}$. We show that a critical point shifts occurs in this case, and obtain an estimate for its intensity.