The dual Derrida-Retaux conjecture

TL;DR

This paper investigates a recursive spin glass model near criticality, confirming a dual conjecture that both the expected value and survival probability decay exponentially with a square-root exponent as the system approaches the critical point.

Contribution

It establishes a dual version of the Derrida-Retaux conjecture, characterizing the decay rates of key quantities in the nearly subcritical regime.

Findings

Both $ ext{E}(X_n)$ and $ ext{P}(X_n eq 0)$ decay exponentially as $p o p_c$.

Decay exponents are proportional to $(p_c - p)^{1/2 + o(1)}$.

Results provide insight into universal behaviors near criticality in hierarchical models.

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 $(p-p_c)^{-\frac12}$ 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 $p \uparrow p_c$.