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### Abstract

We investigate the effect of correlated disorder on the localization transition undergone by a renewal sequence with loop exponent $\alpha >0$, when the correlated sequence is given by another independent renewal set with loop exponent $\widehat{\alpha}>0$. Using the renewal structure of the disorder sequence, we compute the annealed critical point and exponent. Then, using a smoothing inequality for the quenched free energy and second moment estimates for the quenched partition function, combined with decoupling inequalities, we prove that in the case $\widehat{\alpha}>2$ (summable correlations), disorder is irrelevant if $\alpha <1/2$ and relevant if $\alpha >1/2$, which extends the Harris criterion for independent disorder. The case $\widehat{\alpha}\in (1,2)$ (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for $\alpha >1/\widehat{\alpha}$, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case $\widehat{\alpha}\in (0,1)$ is somewhat special but treated for completeness: in this case, disorder has no effect on the quenched free energy, but the annealed model exhibits a phase transition.

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