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The sustainability probability for the critical Derrida–Retaux model

Abstract

We are interested in the recursive model \((Y_n, \, n\ge 0)\) studied by Collet et al. (Commun Math Phys 94:353–370, 1984) and by Derrida and Retaux (J Stat Phys 156:268–290, 2014). We prove that at criticality, the probability \(\mathbf{P}(Y_n>0)\) behaves like \(n^{-2 + o(1)}\) as n goes to infinity; this gives a weaker confirmation of predictions made in Collet et al. (1984), Derrida and Retaux (2014) and Chen et al. (in: Sidoravicius (ed) Sojourns in probability theory and statistical physics-III, Springer, Singapore, 2019). Our method relies on studying the number of pivotal vertices and open paths, combined with a delicate coupling argument.

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Notes

  1. It is crucial to assume that in the Derrida–Retaux model, \(X_0\) is integer-valued; it is an open problem to extend Theorem A without this assumption. See Derrida and Retaux [12] for more discussions.

  2. Degenerate case: when \(|v|=0\), the path from v to v is reduced to the singleton v.

  3. Degenerate case: for \(|v|=0\), the path consisting of the singleton v is considered as open.

  4. We can exchange the order of the sums because \(\sum _{k=0}^\infty \) is a finite sum: \(\mathbf{E}(S_{n+1}^{(k,\ell )}) =0\) for \(k>(n+1)+\ell \).

  5. In Lemma 3.3 below, we are going to work under a stronger integrability assumption, and give dependence in k of the constants \(c_9\) and \(c_{10}\).

  6. The constants are chosen in this order: first \(c_{25}\) then \(c_{23}\), while our presentation is in the opposite order.

  7. It is not a surprise to consider the system at the intermediate generation \(n_0\). A similar idea is going to be used in the proof of the lower bound in Sect. 5.

  8. Since \(m^{Z_M}\, \mathbf{1}_{\{ Z_M \in B_M\} } \le m^{Z_M}\, \mathbf{1}_{\{ Z_M \ge 1\} } \le m^{Y_M}\, \mathbf{1}_{\{ Y_M \ge 1\} }\), it follows from Lemmas 4.4 and 3.1 (iv) that \(\mathbf{E}(m^{Z_M}\, \mathbf{1}_{\{ Z_M \ge 1\} })\) is of order of magnitude \(\frac{1}{M}\). So the meaning of Lemma 4.4 is as follows: if we work with the probability measure having a density proportional to \(m^{Z_M}\), then conditionally on the event \(Z_M\ge 1\), with probability greater than a certain constant (independent of M), \(Z_M\) lies in \(B_M\).

  9. Notation: The letter L is used for the number of open paths for \((Z_n)\), and \({\widetilde{L}}\) for \(({\widetilde{Z}}_n)\).

  10. If there are several such vertices, we can take for example the leftmost one in the lexicographic order.

  11. It is natural to introduce such a decomposition at an intermediate generation M, because the system at generation M has “nicer" properties than at the initial generation, which allows to use some renormalization idea starting from generation M. A similar approach was already used in the proof of the upper bound (Proposition 4.1).

  12. The bridge inequality was stated in [4] for \(N_n^{(0)}\) instead of \(N_n\), but the proof was valid for \(N_n\). Furthermore, \(X_0\) and \(Y_0\) were assumed in [4] to satisfy \({{\mathbb {P}}}(X_0=k) \ge {{\mathbb {P}}}(Y_0=k)\) for all integer \(k\ge 1\) in order to ensure the so-called “XY coupling"; the latter holds automatically in our setting, thanks to the additional assumption \(X_0 \ge Y_0\).

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Acknowledgements

We are indebted to Victor Dagard, Bernard Derrida and Mikhail Lifshits for regular and stimulating discussions on the Derrida–Retaux model throughout the last few years. We also wish to thank two anonymous referees for their careful reading of the original manuscript and for their constructive comments.

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Correspondence to Yueyun Hu.

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Xinxing Chen partially supported by NSFC grants 11771286 and 11531001. Yueyun Hu partially supported by ANR MALIN and ANR SWIWS. Zhan Shi partially supported by ANR MALIN.

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Chen, X., Hu, Y. & Shi, Z. The sustainability probability for the critical Derrida–Retaux model. Probab. Theory Relat. Fields 182, 641–684 (2022). https://doi.org/10.1007/s00440-021-01091-z

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Keywords

  • Derrida–Retaux model
  • Sustainability probability
  • Pivotal vertex
  • Open path

Mathematics Subject Classification

  • 60J80
  • 82B27