The continuous Derrida-Retaux branching process in the Brownian CRT

Abstract

The Derrida-Retaux continuous branching process, which is conjectured to be the scaling limit of the corresponding discrete model and of many other critical hierarchical renormalization models, is a process of cells evolving inhomogeneously on the time interval $[0, 1)$ via linear growth of each cell and independent splitting of cells at a specific rate. In this article, we show that the Derrida-Retaux continuous branching process is, on the one hand, encoded by a family of processes converging to a Brownian motion and on the other hand, fully obtained from the continuum Brownian tree encoded by the limiting Brownian path by time reversal and through length erasure of this tree. This direct representation explains the specific laws governing the significant quantities featuring the model (total mass, number of cells, law of large numbers) and provides a better understanding of its dynamics.