Electronic Journal of Probability

Pinning of a renewal on a quenched renewal

Kenneth S. Alexander and Quentin Berger

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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$.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 6, 48 pp.

Dates
Received: 27 April 2017
Accepted: 3 January 2018
First available in Project Euclid: 12 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1518426053

Digital Object Identifier
doi:10.1214/18-EJP136

Mathematical Reviews number (MathSciNet)
MR3771743

Zentralblatt MATH identifier
1390.60341

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K05: Renewal theory 60K37: Processes in random environments 82B27: Critical phenomena 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
pinning model renewal process quenched disorder localization transition disorder relevance

Rights
Creative Commons Attribution 4.0 International License.

Citation

Alexander, Kenneth S.; Berger, Quentin. Pinning of a renewal on a quenched renewal. Electron. J. Probab. 23 (2018), paper no. 6, 48 pp. doi:10.1214/18-EJP136. https://projecteuclid.org/euclid.ejp/1518426053


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