The KPZ Upper-Tail Field as a KPZ Fixed Point with Brownian Initial Data

Source abstract

The KPZ upper-tail field introduced by Liu and Zhang arises as the scaling limit of the narrow-wedge KPZ fixed point in a microscopic neighborhood of a conditioned large height. In this paper, we give a direct probabilistic identification of the upper-tail field. We identify the grounded positive-time sector with the KPZ fixed point started from the two-sided Brownian $V$-profile $B_{\mathrm{ts}}(2x)-2|x|$, while the negative-time sector is represented through an explicit exponential tilt of the same evolution. The argument uses the directed-landscape variational structure and the local Brownian comparison under upper-tail conditioning. As consequences of these representations, we recover probabilistically the positive- and negative-time large-scale limits of the KPZ upper-tail field.