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Infinite differentiability of the free energy for a Derrida-Retaux system
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abstract
We consider a recursive system which was introduced by Derrida and Retaux (J. Stat. Phys. ${\bf 156}$ (2014) 268-290) as a toy model to study the depinning transition in presence of disorder. Derrida and Retaux predicted the free energy $F_\infty(p)$ of the system exhibit quite an unusual physical phenomenon which is an infinite order phase transition. Hu and Shi (J. Stat. Phys. ${\bf 172}$ (2018) 718-741) studied a special situation and obtained other behavior of the free energy, while insisted on $p=p_c$ being an essential singularity. Recently, Chen, Dagard, Derrida, Hu, Lifshits and Shi (Ann. Probab. ${\bf 49}$ (2021) 637-670) confirmed the Derrida-Retaux conjecture under suitable integrability condition. However, in the mathematical review, it is still unknown whether the free energy is infinitely differentiable at the critical point. So that, we continue to study the infinite differentiability of the free energy in this paper.
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Asymptotic behavior of the generalized Derrida-Retaux recursive model
For a geometric-offspring generalized Derrida-Retaux model, the marginal parameters have explicit asymptotic expansions in every regime, yielding sharp decay rates for the survival probability and mean.
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