An upper tail field of the KPZ fixed point

TL;DR

This paper introduces the upper tail field of the KPZ fixed point, a new limiting random field capturing the behavior near high points, extending the understanding of the KPZ universality class.

Contribution

It constructs and analyzes the upper tail field of the KPZ fixed point, revealing its behavior across full space and connecting it to Brownian fields and the KPZ fixed point.

Findings

The upper tail field exists in full 2D space, unlike the original KPZ fixed point.

In negative time, the upper tail field behaves like a Brownian-type field.

In positive time, it converges to 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.