Renormalization fixed point of the KPZ universality class

TL;DR

This paper introduces a proposed renormalization fixed point for the KPZ universality class, describing evolving interfaces through a random nonlinear semigroup and a variational formula, with connections to the Airy sheet and directed polymers.

Contribution

It proposes a new description of the KPZ fixed point using a random nonlinear semigroup and a variational approach, linking it to the Airy sheet and directed polymers.

Findings

Formulation of the KPZ fixed point via a random nonlinear semigroup.

A variational formula for the transition probabilities.

Connection to the Airy sheet and directed polymers.

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.