Abstract: 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 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 information about the coupling. This difference profile was first studied by Basu, Ganguly, and Hammond, who showed that it is non-decreasing and 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].

Joint work with Shirshendu Ganguly.
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