We are interested in the nearly supercritical regime in a family of max-type recursive models studied by Derrida and Retaux [7], 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 1 2 . This gives a weaker answer to a conjecture of Derrida and Retaux [7]. Other behaviours are obtained when the integrability condition is not satisfied.

We consider a recursive system (X n ) which was introduced by Collet et al. [10]) as a spin glass model, and later by Derrida, Hakim, and Vannimenus [13] and by Derrida and Retaux [14] as a simplified hierarchical renormalization model. The system (X n ) is expected to possess highly nontrivial universalities at or near criticality. In the nearly supercritical regime, Derrida and Retaux [14] conjectured that the free energy of the system decays exponentially with exponent (p − p c ) − 1 2 as p ↓ p c . We study the nearly subcritical regime (p ↑ p c ) and aim at a dual version of the Derrida-Retaux conjecture; our main result states that as n → ∞, both E(X n ) and P(X n = 0) decay exponentially with exponent (p c − p) 1 2 +o(1) , where o(1) → 0 as p ↑ p c .

We show that a coalescence equation exhibits a variety of critical behaviors, depending on the initial condition. This equation was introduced a few years ago to understand a toy model studied by Derrida and Retaux to mimic the depinning transition in presence of disorder. It was shown recently that this toy model exhibits the same critical behaviors as the equation studied in the present work. Here we find several families of exact solutions of this coalescence equation, in particular a family of scaling functions which are closely related to the different possible critical behaviors. These scaling functions lead to new conjectures, in particular on the shapes of the critical trees, that we have checked numerically.