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

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 leads to the question: is there a connection between $\mathcal{D}$ and Brownian local time? Establishing that there is indeed a connection, we prove two results. On a global scale, we show that $\mathcal{D}$ can be written as a Brownian local time patchwork quilt, i.e., as a concatenation of random restrictions of functions which are each absolutely continuous to Brownian local time (of rate four) away from the origin. 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.