Local and global comparisons of the Airy difference profile to Brownian local time

TL;DR

This paper investigates the difference profile of the Airy sheet, revealing its fractal structure and showing it is globally absolutely continuous with Brownian local time, thus connecting complex stochastic processes with classical Brownian motion.

Contribution

It establishes the absolute continuity of the Airy difference profile to Brownian local time and provides explicit local limits, advancing understanding of the Airy sheet's fractal structure.

Findings

The difference profile is absolutely continuous to Brownian local time.

The difference profile's points of increase relate to Brownian local time.

Explicit local limits of the difference profile are obtained at typical points.

Abstract

There has recently been much activity within the Kardar-Parisi-Zhang universality class spurred by the construction of the canonical limiting object, the parabolic Airy sheet $\mathcal{S}:\mathbb{R}^2\to\mathbb{R}$ [arXiv:1812.00309]. The parabolic Airy sheet provides a coupling of parabolic Airy$_2$ processes -- a universal limiting geodesic weight profile in planar last passage percolation models -- and a natural goal is to understand this coupling. Geodesic geometry suggests that the difference of two parabolic Airy$_2$ processes, i.e., a difference profile, encodes important structural information. This difference profile $\mathcal{D}$, given by $\mathbb{R}\to\mathbb{R}:x\mapsto \mathcal{S}(1,x)-\mathcal{S}(-1,x)$, was first studied by Basu, Ganguly, and Hammond [arXiv:1904.01717], who showed that it is monotone and almost everywhere constant, with its points of non-constancy forming a set of Hausdorff dimension $1/2$. Noticing that this is also the Hausdorff dimension of the zero set of Brownian motion, we adopt a different approach. Establishing previously inaccessible fractal structure of $\mathcal{D}$, we prove, on a global scale, that $\mathcal{D}$ is absolutely continuous on compact sets to Brownian local time (of rate four) in the sense of increments, which also yields the main result of [arXiv:1904.01717] as a simple corollary. Further, on a local scale, we explicitly obtain Brownian local time of rate four as a local limit of $\mathcal{D}$ at a point of increase, picked by a number of methods, including at a typical point sampled according to the distribution function $\mathcal{D}$. Our arguments rely on the representation of $\mathcal{S}$ in terms of a last passage problem through the parabolic Airy line ensemble and an understanding of geodesic geometry at deterministic and random times.