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The Derrida-Retaux model on a geometric Galton-Watson tree

Gerold Alsmeyer, Yueyun Hu, Bastien Mallein

Abstract

We consider a generalized Derrida-Retaux model on a Galton-Watson tree with a geometric offspring distribution. For a class of recursive systems, including the Derrida-Retaux model with either a geometric or exponential initial distribution, we characterize the critical curve using an involution-type equation and prove that the free energy satisfies the Derrida-Retaux conjecture.

The Derrida-Retaux model on a geometric Galton-Watson tree

Abstract

We consider a generalized Derrida-Retaux model on a Galton-Watson tree with a geometric offspring distribution. For a class of recursive systems, including the Derrida-Retaux model with either a geometric or exponential initial distribution, we characterize the critical curve using an involution-type equation and prove that the free energy satisfies the Derrida-Retaux conjecture.

Paper Structure

This paper contains 9 sections, 21 theorems, 214 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Solvable discrete-time DR models with two parameters
  3. Generalized DR model on Z+ with linear fractional input
  4. Generalized DR model on R+ with continuous linear fractional input
  5. Simple properties of the Derrida-Retaux recursion
  6. Evolution along the critical line
  7. Functional equation and analysis of the critical curve
  8. The critical regime
  9. The Derrida-Retaux conjecture

Key Result

Theorem 1.1

Under the assumptions ass:psi, the function $h$ is nonincreasing, Lipschitz continuous and satisfies $0\le h(x)\le (-x)_+$ for all $x\in\mathbb{R}$. Moreover, $h$ is the unique nonzero solution to the functional equation Finally, the domains $\mathcal{P},\mathcal{C}$ and $\mathcal{U}$ can be characterized as follows:

Figures (2)

  • Figure 1: Decomposition of the phase space for a function $\Psi$ defined by $\Psi(x)= x^2/{2(1+x-\sqrt{1+2x})}$ for $x \in [-0.5,0.5]$, extended to $\mathbb{R}$ in such a way that \ref{['ass:psi']} holds. For this function, the critical curve $h$ is given by $x \mapsto x^2/2$ on $[-0.5,0.5]$ and drawn in red. Additionally, slightly supercritical and subcritical trajectories for $(u,v)$ are depicted.
  • Figure 2: Numerical computations of $g_{n}(x)-x$ and $x^2/2$, where $\eta=-0.5, K=10,$ and $\Psi(x)= \frac{1+2x}{1+x}$ corresponding to the generalized DR model described in Section \ref{['def-LF']} with ${\tt Z}=1$ and $p=0.5$.

Theorems & Definitions (46)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Linear fractional distribution
  • proof
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 36 more