Disorder and critical phenomena: the \(\alpha =0\) copolymer model

  • Quentin Berger
  • Giambattista GiacominEmail author
  • Hubert Lacoin


The 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 \(1+ \alpha \;\geqslant \;1\). 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 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.


Copolymer model Phase transition Critical phenomena Influence of disorder Strong disorder renormalization group 

Mathematics Subject Classification

60K37 82B27 82B44 60K35 82D60 



This work has been performed in part when two of the authors, G.G. and H.L., were at the Institut Henri Poincaré (2017, spring-summer trimester) and we thank IHP for the hospitality. The visit to IHP by H.L. was supported by the Fondation de Sciences Mathématiques de Paris. G.G. acknowledges the support of grant ANR-15-CE40-0020. H.L. acknowledges the support of a productivity grant from CNPq and a Grant Jovem Cientśta do Nosso Estado from FAPERJ.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Probabilités, Statistique et Modélisation, UMR 8001Sorbonne UniversitéParisFrance
  2. 2.Sorbonne Paris Cité, Laboratoire de Probabilités, Statistique et Modélisation, UMR 8001Université Paris DiderotParisFrance
  3. 3.IMPA-Instituto de Matemática Pura e AplicadaRio de JaneiroBrazil

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