Pinning of a renewal on a quenched renewal

Abstract

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 $\exp (\beta \sum _{n=1}^N{\mathbf{1}}_{\{{\sigma }_n\in \tau \}})$. Our assumption is that $\tau$ and $\sigma$ have gap distributions with power-law-decay exponents $1+\alpha$ and $1+\tilde{\alpha }$ respectively, with $\alpha \ge 0,\tilde{\alpha }>0$. 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 ${\beta }_c^{\mathrm{ann}}$. We show that ${\beta }_c={\beta }_c^{\mathrm{ann}}$ whenever $\alpha +\tilde{\alpha }\ge 1$, and ${\beta }_c=0$ if and only if the renewal $\tau \cap \sigma$ is recurrent. On the other hand, we show ${\beta }_c>{\beta }_c^{\mathrm{ann}}$ when $\alpha +\frac{3}{2}\tilde{\alpha }<1$. We give evidence that this should in fact be true whenever $\alpha +\tilde{\alpha }<1$, providing examples for all such $\alpha ,\tilde{\alpha }$ of distributions of $\tau ,\sigma$ for which ${\beta }_c>{\beta }_c^{\mathrm{ann}}$. 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 $\alpha >0$.