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 S : R2 & RARR; R (Dauvergne, Ortmann and Virag (2018)). The parabolic Airy sheet provides a coupling of parabolic Airy2 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 Airy2 processes, i.e., a difference profile, encodes important structural information. This difference profile D, given by R & RARR; R : x & RARR; S(1, x) - S(-1, x), was first studied by Basu, Ganguly and Hammond (2019), 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 D, we prove, on a global scale, that 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 Basu, Ganguly and Hammond (2019) as a simple corollary. Further, on a local scale, we explicitly obtain Brownian local time of rate four as a local limit of D at a point of increase, picked by a number of methods, including at a typical point sampled according to the distribution function D. Our arguments rely on the representation of 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.