The Airy difference profile and Brownian local time
There has recently been much activity within the Kardar-Parisi-Zhang universality class spurred by the construction of a canonical limiting object, the parabolic Airy sheet, by Dauvergne-Ortmann-Virág [DOV]. 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 question is to understand this coupling. Geodesic geometry suggests that the difference of two parabolic Airy_2 processes, i.e., a difference profile, encodes important information about the coupling. This difference profile was first studied by Basu, Ganguly, and Hammond, who showed that it is almost everywhere constant, with its points of non-constancy forming a set of Hausdorff dimension 1/2. This also being the Hausdorff dimension of the zero set of Brownian motion, we are led to ask: is there a connection between the two objects? This talk will elucidate such a connection on both local and global scales, making use of the representation of the parabolic Airy sheet via a continuous counterpart of the RSK correspondence as introduced in [DOV].
This is based on joint work with Shirshendu Ganguly.