Pinning of a renewal on a quenched renewal

We introduce the pinning model on a quenched renewal, which is an instance of a (strongly correlated) disordered pinning model. The potential takes value 1 at the renewal times of a quenched realization of a renewal process σ, and 0 elsewhere, so nonzero potential values become sparse if the gaps in σ have infinite mean. The " polymer " – of length σ_N – is given by another renewal τ , whose law is modified by the Boltzmann weight exp(\beta\sum_{n=1}^N 1_{\sigma_n \in \tau}). Our assumption is that τ and σ have gap distributions with power-law-decay exponents 1 + α and 1 + \tilde α respectively, with α ≥ 0, \tilde α > 0. There is a localization phase transition: above a critical value β_c the free energy is positive, meaning that τ is pinned on the quenched renewal σ. We consider the question of relevance of the disorder, that is to know when β c differs from its annealed counterpart β^{ann}_c. We show that β_c = β^{ann}_c whenever α + \tilde α ≥ 1, and β_c = 0 if and only if the renewal τ ∩ σ is recurrent. On the other hand, we show β_c > β^{ann}_c when α + 3 \tilde α/2 β^{ann}_c. We additionally consider two natural variants of the model: one in which the polymer and disorder are constrained to have equal numbers of renewals (σ_N = τ_N), and one in which the polymer length is τ_N rather than σ_N. In both cases we show the critical point is the same as in the original model, at least when α > 0.