Renormalization Fixed Point of the KPZ Universality Class

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 Airy[subscript 2] 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.National Science Foundation (U.S.) (PIRE Grant OISE-07-30136 and DMS-1208998)Microsoft Research (Schramm Memorial Fellowship)Clay Mathematics Institute (Clay Research Fellowship)Institut Henri Poincare (Poincare Chair)David & Lucile Packard Foundation (Packard Fellowships for Science and Engineering) Remove selecte