The Critical 2D KPZ Process
Abstract
This work constructs a non-trivial solution to the ill-defined two-dimensional Kardar-Parisi-Zhang (KPZ) equation at criticality: a distribution-valued stochastic process on the plane, which we call the Critical 2D KPZ Process. It arises as the limit in law of centred log partition functions of two-dimensional Gaussian directed polymers at criticality, viewed as discretised KPZ solutions. It is also a measurable function of the Critical 2D Stochastic Heat Flow (SHF) via a renormalised Cole-Hopf transformation, obtained as the limit of the centred logarithm of the spatially mollified SHF as the smoothing is removed. The Critical 2D KPZ process is non-Gaussian and its covariance displays a log-log blow-up near the diagonal. Gaussian fluctuations emerge when averaging on shrinking spatial scales. The results presented here were obtained through extensive interactions with generative AI (ChatGPT 5.5, 5.6 and 6); see Appendix C for more details. The proof builds on key recent and older advances in our field, combined in novel ways with substantial AI input. We view this manuscript as a "community project" that represents more than a decade's collective effort. We have worked on making the presentation as accessible as possible, highlighting the main ideas and their relations to the literature.
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BibTeX
@misc{berger2026critical,
title = {{The Critical 2D KPZ Process}},
author = {Quentin Berger and Francesco Caravenna and Jeremy Clark and Alexander Dunlap and Yu Gu and Ziyang Liu and Shuta Nakajima and Rongfeng Sun and Li-Cheng Tsai and Zuodi Xie and Nikos Zygouras},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31614},
eprint = {2610.31614},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31614},
}
RIS
TY - UNPB
TI - The Critical 2D KPZ Process
AU - Quentin Berger
AU - Francesco Caravenna
AU - Jeremy Clark
AU - Alexander Dunlap
AU - Yu Gu
AU - Ziyang Liu
AU - Shuta Nakajima
AU - Rongfeng Sun
AU - Li-Cheng Tsai
AU - Zuodi Xie
AU - Nikos Zygouras
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31614
UR - https://arxiv.org/abs/2610.31614
AB - This work constructs a non-trivial solution to the ill-defined two-dimensional Kardar-Parisi-Zhang (KPZ) equation at criticality: a distribution-valued stochastic process on the plane, which we call the Critical 2D KPZ Process. It arises as the limit in law of centred log partition functions of two-dimensional Gaussian directed polymers at criticality, viewed as discretised KPZ solutions. It is also a measurable function of the Critical 2D Stochastic Heat Flow (SHF) via a renormalised Cole-Hopf transformation, obtained as the limit of the centred logarithm of the spatially mollified SHF as the smoothing is removed. The Critical 2D KPZ process is non-Gaussian and its covariance displays a log-log blow-up near the diagonal. Gaussian fluctuations emerge when averaging on shrinking spatial scales. The results presented here were obtained through extensive interactions with generative AI (ChatGPT 5.5, 5.6 and 6); see Appendix C for more details. The proof builds on key recent and older advances in our field, combined in novel ways with substantial AI input. We view this manuscript as a "community project" that represents more than a decade's collective effort. We have worked on making the presentation as accessible as possible, highlighting the main ideas and their relations to the literature.
ER -