Renormalization fixed point of the KPZ universality class
Abstract
The one dimensional Kardar-Parisi-Zhang universality class is believed to describe many types of evolving interfaces which have the same characteristic scaling exponents. These exponents lead to a natural renormalization/rescaling on the space of such evolving interfaces. We introduce and describe the renormalization fixed point of the Kardar-Parisi-Zhang universality class in terms of a random nonlinear semigroup with stationary independent increments, and via a variational formula. Furthermore, we compute a plausible formula the exact transition probabilities using replica Bethe ansatz. The semigroup is constructed from the Airy sheet, a four parameter space-time field which is the Airy2 process in each of its two spatial coordinates. Minimizing paths through this field describe the renormalization group fixed point of directed polymers in a random potential. At present, the results we provide do not have mathematically rigorous proofs, and they should at most be considered proposals.
Cite
@article{arxiv.1103.3422,
title = {Renormalization fixed point of the KPZ universality class},
author = {Ivan Corwin and Jeremy Quastel and Daniel Remenik},
journal= {arXiv preprint arXiv:1103.3422},
year = {2020}
}
Comments
19 pages, 2 figures; updated and expanded version with addition author (Remenik)