Critical moment-generating-function asymptotics conjecture for the Derrida–Retaux recursion

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

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

Critical generating-function asymptotics conjecture. As $n\to\infty$, for $s\in(0,m)$ with $s\ne1$,

$${\bf E}(s^{X_n})-1\sim\frac{4m}{(m-1)^2}\frac{s-1}{m-s}\frac{1}{n^2},$$

and

$${\bf E}(m^{X_n})-1\sim\frac{2}{m-1}\frac{1}{n}.$$

These estimates describe the rate at which the moment-generating function approaches its critical limiting value and complement the conjectured moment asymptotics. They remain conjectural in the paper.