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Pré-Publication, Document De Travail Année : 2023

The dual Derrida--Retaux conjecture

Résumé

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 $p \uparrow p_c$.
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Dates et versions

hal-04133340 , version 1 (19-06-2023)

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Xinxing Chen, Yueyun Hu, Zhan Shi. The dual Derrida--Retaux conjecture. 2023. ⟨hal-04133340⟩
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