Hierarchical pinning model with site disorder : disorder is marginally relevant
Résumé
We study a hierarchical disordered pinning model with site disorder for which, like in the bond disordered case (Derrida et al. in J Stat Phys 66: 1189-1213, 1992 and Giacomin et al. in Probab. Theor. Rel. Fields 2009, arXiv: 0711.4649 [math.PR]), there exists a value of a parameter b (which enters in the definition of the hierarchical lattice) that separates an irrelevant disorder regime and a relevant disorder regime. We show that for such a value of b the critical point of the disordered system is different from the critical point of the annealed version of the model. The proof goes beyond the technique used in Giacomin et al. (Probab. Theor. Rel. Fields 2009, arXiv: 0711.4649 [math.PR]) and it takes explicitly advantage of the inhomogeneous character of the Green function of the model.