Scaling limit of the continuous Derrida–Retaux branching process

Determine whether the continuous Derrida–Retaux branching process is the scaling limit of the discrete Derrida–Retaux renormalization model and of other critical hierarchical renormalization models.

Background

The paper studies the continuous Derrida–Retaux branching process, a population model in which cells grow linearly and split at a mass-dependent rate. The authors represent this process through a Brownian continuum random tree and derive structural and asymptotic properties, including laws for total mass, cell counts, and empirical cell masses.

The relationship between this continuous process and discrete critical hierarchical renormalization models is not established in the paper. In particular, the authors identify as a conjecture the claim that the continuous Derrida–Retaux branching process arises as the scaling limit of the corresponding discrete Derrida–Retaux model, as well as of more general critical hierarchical renormalization models. The paper instead focuses on the Brownian continuum-tree representation and does not prove this discrete-to-continuous convergence.