# Disorder and critical phenomena: the $\alpha=0$ copolymer model

Abstract : The generalized copolymer model is a disordered system built on a discrete renewal process with inter-arrival distribution that decays in a regularly varying fashion with exponent $1+ \alpha\geq 1$. It exhibits a localization transition which can be characterized in terms of the free energy of the model: the free energy is zero in the delocalized phase and it is positive in the localized phase. This transition, which is observed when tuning the mean $h$ of the disorder variable, has been tackled in the physics literature notably via a renormalization group procedure that goes under the name of \emph{strong disorder renormalization}. We focus on the case $\alpha=0$ -- the critical value $h_c(\beta)$ of the parameter $h$ is exactly known (for every strength $\beta$ of the disorder) in this case -- and we provide precise estimates on the critical behavior. Our results confirm the strong disorder renormalization group prediction that the transition is of infinite order, namely that when $h\searrow h_c(\beta)$ the free energy vanishes faster than any power of $h-h_c(\beta)$. But we show that the free energy vanishes much faster than the physicists' prediction.
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Type de document :
Pré-publication, Document de travail
26 pages, 1 figure. 2017
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Littérature citée [34 références]

https://hal.archives-ouvertes.fr/hal-01659845
Contributeur : Quentin Berger <>
Soumis le : vendredi 8 décembre 2017 - 20:35:58
Dernière modification le : lundi 5 mars 2018 - 20:40:38

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BGL-alpha0.pdf
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• HAL Id : hal-01659845, version 1
• ARXIV : 1712.02261

### Citation

Quentin Berger, Giambattista Giacomin, Hubert Lacoin. Disorder and critical phenomena: the $\alpha=0$ copolymer model. 26 pages, 1 figure. 2017. 〈hal-01659845〉

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