  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 $(p-p_c)^{-\frac12}$ as $p \downarrow p_c$. We
study the nearly subcritical regime ($p \uparrow p_c$) and aim at a dual
version of the Derrida-Retaux conjecture; our main result states that as $n \to
\infty$, both $\E(X_n)$ and $\P(X_n\neq 0)$ decay exponentially with exponent
$(p_c-p)^{\frac12 +o(1)}$, where $o(1) \to 0$ as $p \uparrow p_c$.
