Critical conditional geometric-limit conjecture for the Derrida–Retaux recursion

Let $m$ be the recursion parameter and let $X_0$ be integer-valued. Assume the critical-regime condition

$$(m-1){\bf E}(X_0m^{X_0})={\bf E}(m^{X_0})<\infty.$$

Conditional geometric-limit conjecture. Conditionally on $X_n\ne0$, the random variable $X_n$ converges weakly to a random variable $Y_\infty$ satisfying

$${\bf P}(Y_\infty=k)=\frac{m-1}{m^k},\qquad k\ge1.$$

This predicts the limiting distribution of the positive state at criticality. It is the second part of the paper’s proposed asymptotic picture, and the authors had not succeeded in making the underlying arguments rigorous.