The stable Derrida--Retaux system at criticality

TL;DR

This paper studies the critical behavior of the Derrida--Retaux recursive system, providing a unified approach to analyze its phase transition and universality properties, especially when initial conditions do not satisfy certain integrability assumptions.

Contribution

It introduces a unified method for systems under a domination condition and characterizes the large-time behavior at criticality without the integrability assumption.

Findings

Established an upper bound for derivatives of the moment generating function.

Identified the order of magnitude of the product of moment generating functions at criticality.

Confirmed and extended previous results on the system's phase transition.

Abstract

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).