Journal of Statistical Physics

, Volume 172, Issue 3, pp 718–741 | Cite as

The Free Energy in the Derrida–Retaux Recursive Model

  • Yueyun HuEmail author
  • Zhan Shi


We are interested in a simple max-type recursive model studied by Derrida and Retaux (J Stat Phys 156:268–290, 2014) in the context of a physics problem, and find a wide range for the exponent in the free energy in the nearly supercritical regime.


Max-type recursive model Free energy Multi-scale analysis 

Mathematics Subject Classification

60J80 82B44 



We are grateful to Bernard Derrida who introduced us to the problem, and with whom we have had regular discussions for two years. We wish to thank Nina Gantert for many discussions, Quentin Berger for enlightenment on renormalisation models, and Chunhua Ma, Bastien Mallein and Quan Shi for pointing out [20] to us. Two anonymous referees have carefully read the manuscript; their insightful comments have led to improvements in the paper. The project was partly supported by ANR MALIN (ANR-16-CE93-0003); Y.H. also acknowledges support from ANR SWiWS (ANR-17-CE40-0032).


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LAGAUniversité Paris XIIIVilletaneuseFrance
  2. 2.LPMAUniversité Pierre et Marie CurieParis Cedex 05France

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