Abstract
We are interested in the nearly supercritical regime in a family of max-type recursive models studied by Collet, Eckman, Glaser and Martin (Comm. Math. Phys. 94 (1984) 353–370) and by Derrida and Retaux (J. Stat. Phys. 156 (2014) 268–290) and prove that, under a suitable integrability assumption on the initial distribution, the free energy vanishes at the transition with an essential singularity with exponent . This gives a weaker answer to a conjecture of Derrida and Retaux (J. Stat. Phys. 156 (2014) 268–290). Other behaviours are obtained when the integrability condition is not satisfied.
Acknowledgements
We are grateful to two anonymous referees whose insightful comments have led to improvements in the presentation of the paper.
The first author was partially supported by NSFC grants 11771286 and 11531001.
The second, third and sixth authors were partially supported by ANR project MALIN 16-CE93-0003.
The fourth author was partially supported by ANR project MALIN 16-CE93-0003 and by ANR project SWIWS 17-CE40-0032-02.
The fifth author was partially supported by RFBR-DFG grant 20-51-12004.
Citation
Xinxing Chen. Victor Dagard. Bernard Derrida. Yueyun Hu. Mikhail Lifshits. Zhan Shi. "The Derrida–Retaux conjecture on recursive models." Ann. Probab. 49 (2) 637 - 670, March 2021. https://doi.org/10.1214/20-AOP1457
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