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We investigate disorder relevance for the pinning of a renewal whoseinter-arrival law has tail exponent $\alpha>0$ when the law of the randomenvironment is in the domain of attraction of a stable law with parameter$\gamma \in (1,2)$. We prove that in this case, the effect of disorder is notdecided by the sign of the specific heat exponent as predicted by Harriscriterion but that a new criterion emerges to decide disorder relevance. Moreprecisely we show that when $\alpha>1-\gamma^{-1}$ there is a shift of thecritical point at every temperature whereas when $\alpha< 1-\gamma^{-1}$, athigh temperature the quenched and annealed critical point coincide, and thecritical exponents are identical.

Disorder relevance without Harris Criterion: the case of pinning model with $\gamma$-stable environment