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An exactly solvable continuous-time Derrida-Retaux model. (English) Zbl 1439.82018

B. Derrida and M. Retaux [J. Stat. Phys. 156, No. 2, 268–290 (2014; Zbl 1312.82005)] introduced a max-type recursive model to represents a simplified version of the problem of the depinning transition in the limit of strong disorder. This max-type recursive model is defined by \[X_{n+1}\overset{(d)}{=}(X_n+\widetilde{X}_{n}-1)_+,\ \text{ for all } n\geq 0,\] where, for any \(x\in \mathbb{R}\), \(x_{+} := \max(x, 0),\) \(\widetilde{X}_n\) denotes an independent copy of \(X_n\), and \(\overset{(d)}{=}\) stands for the identity in distribution. In this present paper, the authors study an exactly solvable version of a continuous-time generalization of the Derrida and Retaux (DR) model. A process description of this model is given. The painting scheme on a Yule tree is illustrated. The definition of a continuous-time DR tree \((\mu_t)_{t\geq0}\) is presented. The authors obtain preliminary results on the asymptotic behavior of a general continuous-time DR model. The authors give a classification of this DR model in pinned, unpinned and critical behaviors for any starting distribution. The pinned and unpinned regions of the solvable DR model are plotted. Also, the phase diagram of the differential system is plotted. The critical regime of the model is rigorously studied. Some quantitative characteristics of the conditioned (red) tree are presented. In the last section, the authors give some results on the continuous-time DR model with a general initial distribution on \(\mathbb{R}_+\). Some open problems are presented.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
05C05 Trees

Citations:

Zbl 1312.82005
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References:

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