  We consider the hierarchical disordered pinning model studied in [9], which
exhibits a localization/delocalization phase transition. In the case where the
disorder is i.i.d. (independent and identically distributed), the question of
relevance/irrelevance of disorder (i.e. whether disorder changes or not the
critical properties with respect to the homogeneous case) is by now
mathematically rather well understood [14,15]. Here we consider the case where
randomness is spatially correlated and correlations respect the hierarchical
structure of the model; in the non-hierarchical model our choice would
correspond to a power-law decay of correlations. In terms of the critical
exponent of the homogeneous model and of the correlation decay exponent, we
identify three regions. In the first one (non-summable correlations) the phase
transition disappears. In the second one (correlations decaying fast enough)
the system behaves essentially like in the i.i.d. setting and the
relevance/irrelevance criterion is not modified. Finally, there is a region
where the presence of correlations changes the critical properties of the
annealed system.
