Hierarchical pinning model : low disorder relevance in the b = s case

We consider a hierarchical pinning model introduced by B. Derrida, V. Hakim and J. Vannimenus which undergoes a localization/delocalization phase transition. This model depends on two parameters $b$ and $s$. We show that in the particular case where $b \eq s$, the disorder is weakly relevant, in the sense that at any given temperature, the quenched and the annealed critical points coincide. This is in contrast with the case where $b \neq s$. Keywords: Hierarchical Models, Quadratic Recurrence Equations, Pinning Models Mathematics