The critical two-dimensional KPZ process

At the critical disorder scale for a two-dimensional directed polymer, do the centred logarithms of its partition functions converge to a well-defined random distribution? The Gaussian disorder model, scaling window and topology are part of the question.

Why it matters and what counts as progress

Why it matters: Critical stochastic growth links random interfaces, directed polymers and singular stochastic equations. A precise limiting object distinguishes critical behaviour from subcritical Gaussian limits.

What counts as progress: Construct and characterize the critical limit, prove convergence under stated initial data, or extend its universality and regularity. Distinguish this model-specific limit from a result for every two-dimensional growth system.

How the problem developed

1986 · Research result [1]
Mehran Kardar, Giorgio Parisi and Yi-Cheng Zhang Introduced the stochastic-growth equation and scaling perspective behind the KPZ universality class.
2023 · Research result [1]
Francesco Caravenna, Rongfeng Sun and Nikos Zygouras Constructed the critical 2D stochastic heat flow, the object from which the later logarithmic field is built.
22 Oct 2026 · AI contribution [1]
Berger and ten coauthors with ChatGPT Reported convergence to the critical 2D KPZ process and non-Gaussianity. The authors explain surviving model-generated arguments and their own structural guidance.
28 Oct 2026 · Research result [2][3]
Scott Armstrong Registered corrected Lean counterparts and a detailed source correspondence. His separate four-result extension is distinct from the original AI contribution.

AI contribution

The Critical 2D KPZ Process

Quentin Berger, Francesco Caravenna, Jeremy Clark, Alexander Dunlap, Yu Gu, Ziyang Liu, Shuta Nakajima, Rongfeng Sun, Li-Cheng Tsai, Zuodi Xie, Nikos Zygouras

For directed polymers on Z² in a Gaussian environment in the specified critical window with fixed parameter ϑ, centred log-partition fields converge to a uniquely constructed critical 2D KPZ random distribution, jointly for finitely many times and initial data in the stated local negative Sobolev topology. For flat initial data the limit is non-Gaussian, despite Gaussian small-scale averages.

AI involvementAI-centered

ImportanceNotable

Technical verificationPartial

Human understandingFull