Last passage isometries for the directed landscape

TL;DR

This paper establishes a last passage isometry for the directed landscape on specific sets, linking it to Brownian functions, and explores implications for the Airy profile, KPZ fixed point, and geodesic shapes.

Contribution

It introduces a novel last passage isometry for the directed landscape on certain sets, connecting it to Brownian functions and extended landscape marginals.

Findings

Airy difference profile is locally absolutely continuous with respect to Brownian local time

KPZ fixed point from two narrow wedges has a Brownian-Bessel decomposition

Directed landscape is determined by its geodesic shapes

Abstract

Consider the restriction of the directed landscape $\mathcal L(x, s; y, t)$ to a set of the form $\{x_1, \dots, x_k\} \times \{s_0\} \times \mathbb R \times \{t_0\}$. We show that on any such set, the directed landscape is given by a last passage problem across $k$ locally Brownian functions. The $k$ functions in this last passage isometry are built from certain marginals of the extended directed landscape. As applications of this construction, we show that the Airy difference profile is locally absolutely continuous with respect to Brownian local time, that the KPZ fixed point started from two narrow wedges has a Brownian-Bessel decomposition around its cusp point, and that the directed landscape is a function of its geodesic shapes.