The Derrida–Retaux conjecture on recursive models

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 $\frac{1}{2}$. 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.