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 sigma, and 0 elsewhere, so nonzero potential values become sparse if the gaps in sigma have infinite mean. The "polymer" - of length sigma(N) - is given by another renewal tau, whose law is modified by the Boltzmann weight exp(beta Sigma(N)(n=1) 1{sigma(n) subset of tau}). Our assumption is that tau and sigma have gap distributions with power-law-decay exponents 1 + alpha and 1 + (alpha) over tilde respectively, with alpha >= 0; (alpha) over tilde > 0. There is a localization phase transition: above a critical value beta(c) the free energy is positive, meaning that tau is pinned on the quenched renewal sigma. We consider the question of relevance of the disorder, that is to know when beta(c) differs from its annealed counterpart beta(ann)(c). We show that beta(c) = beta(ann)(c) whenever alpha + (alpha) over tilde >= 1, and beta(c) = 0 if and only if the renewal tau boolean AND sigma is recurrent. On the other hand, we show beta(c) > beta(ann)(c) when alpha + 3/2 (alpha) over tilde < 1. We give evidence that this should in fact be true whenever alpha + <(alpha)over tilde> < 1, providing examples for all such alpha, <(alpha)over tilde> of distributions of tau, sigma for which beta(c) > beta(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 (sigma(N) = tau(N)), and one in which the polymer length is tau(N) rather than sigma(N). In both cases we show the critical point is the same as in the original model, at least when alpha > 0.