An upper tail field of the KPZ fixed point

Abstract

The KPZ fixed point is a (1+1)-dimensional space-time random field conjectured to be the universal limit for models within the Kardar-Parisi-Zhang (KPZ) universality class. We consider the KPZ fixed point with the narrow-wedge initial condition, conditioning on a large value at a specific point. By zooming in the neighborhood of this high point appropriately, we obtain a limiting random field, which we call an upper tail field of the KPZ fixed point. Different from the KPZ fixed point, where the time parameter has to be nonnegative, the upper tail field is defined in the full $2$-dimensional space. Especially, if we zoom out the upper tail field appropriately, it behaves like a Brownian-type field in the negative time regime, and the KPZ fixed point in the positive time regime. One main ingredient of the proof is an upper tail estimate of the joint tail probability functions of the KPZ fixed point near the given point, which generalizes the well known one-point upper tail estimate of the GUE Tracy-Widom distribution.