For the discrete Derrida–Retaux models in the critical regime, after the model-specific rescaling of time and state, the resulting processes converge in distribution, in the relevant process topology, to the continuous Derrida–Retaux branching process on the time interval [0,1).

Derrida–Retaux scaling-limit conjecture

The conjecture says that suitably rescaled critical discrete Derrida–Retaux models converge to a universal continuous branching process. No source identifies an original proposer or date.

Known results

An exactly solvable continuous-time Derrida–Retaux model has a process-level scaling limit, but this does not prove the discrete-model conjecture (2018).
A generalized discrete model with renewal rate $\alpha=a/k$ has weak process convergence to a generalized continuous-time model, but not in the stated critical regime (November 2024).

September 10, 2026 Brownian-CRT representation

Thomas Duquesne and Zhan Shi’s paper represents the proposed continuous limit using Brownian motion and the Brownian continuum random tree, and derives further properties of it. The paper calls discrete-to-continuous convergence a conjecture; this reported advance is therefore unverified as a solution.

Current status (as of September 2026): The discrete-to-continuous scaling-limit conjecture remains open; the candidate continuous limit has a new Brownian-CRT representation, but convergence from the discrete models is unproved.

Derrida–Retaux scaling-limit conjecture

Primary source

Additional references

Progress summary

Solutions 0