The dual Derrida-Retaux conjecture

Abstract

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 $(p-p_c{)}^{-\frac{1}{2}}$ 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 $\P (X_n\ne 0)$ decay exponentially with exponent $(p_c-p{)}^{\frac{1}{2}+o(1)}$, where $o(1)\to 0$ as $p\uparrow p_c$.