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 $\alpha \geq 0,\tilde \alpha >0$. 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 \geq 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 32\, \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$.