A conditional scaling limit of the KPZ fixed point with height tending to infinity at one location

TL;DR

This paper studies the asymptotic behavior of the KPZ fixed point conditioned on large height at a point, revealing limit processes related to Brownian bridges and providing insights into geodesic behaviors in the directed landscape.

Contribution

It establishes a conditional limit theorem for KPZ fluctuations with height tending to infinity, characterizing the limit as a functional of Brownian bridges and dependent on initial conditions.

Findings

Limit process for KPZ with step initial condition is the minimum of two Brownian bridges.

Limit process for flat initial condition involves the minimum of two Brownian bridges plus a Gaussian variable.

Provides asymptotic behavior of point-to-point geodesics conditioned on their length.

Abstract

We consider the asymptotic behavior of the KPZ fixed point $\{\mathsf H(x,t)\}_{x\in\mathbb R, t>0}$ conditioned on $\mathsf H(0,T)=L$ as $L$ goes to infinity. The main result is a conditional limit theorem for the fluctuations of $\mathsf H$ in the region near the line segment connecting the origin $(0,0)$ and $(0,T)$ for both step and flat initial conditions. The limit random field can be represented as a functional of two independent Brownian bridges, and in addition the limit random field depends also on the initial law of the KPZ fixed point. In particular for temporal fluctuations, the limit process indexed by line segment between $(0,0)$ and $(0,T)$, when the KPZ is with step initial condition, has the law of the minimum of two independent Brownian bridges; and when the KPZ is with flat initial condition the limit process has the law of the minimum of two independent Brownian bridges, each in addition perturbed by a common Gaussian random variable. For spatial-temporal fluctuations, the conditional limit theorem sheds light on the asymptotic behaviors of the point-to-point geodesic of the directed landscape conditioned on its length and as the length tends to infinity.