  The Derrida--Retaux recursive system was investigated by Derrida and Retaux
(2014) as a hierarchical renormalization model in statistical physics. A
prediction of Derrida and Retaux (2014) on the free energy has recently been
rigorously proved (Chen, Dagard, Derrida, Hu, Lifshits and Shi (2019+)),
confirming the Berezinskii--Kosterlitz--Thouless-type phase transition in the
system. Interestingly, it has been established in Chen, Dagard, Derrida, Hu,
Lifshits and Shi (2019+) that the prediction is valid only under a certain
integrability assumption on the initial distribution, and a new type of
universality result has been shown when this integrability assumption is not
satisfied. We present a unified approach for systems satisfying a certain
domination condition, and give an upper bound for derivatives of all orders of
the moment generating function. When the integrability assumption is not
satisfied, our result allows to identify the large-time order of magnitude of
the product of the moment generating functions at criticality, confirming and
completing a previous result in Collet, Eckmann, Glaser and Martin (1984).
