The Derrida–Retaux conjecture on recursive models

We are interested in the nearly supercritical regime a family of max-type recursive models studied by Collet, Eckman, Glaser and Martin (Comm. Math. Phys. 94 (1984) 353–370) Derrida Retaux (J. Stat. 156 (2014) 268–290) prove that, under suitable integrability assumption on initial distribution, free energy vanishes at transition with an essential singularity exponent 12. This gives weaker answer to conjecture 268–290). Other behaviours obtained when condition is not satisfied.