Disjoint optimizers and the directed landscape
arXiv:2102.00954
Abstract
We study maximal length collections of disjoint paths, or 'disjoint optimizers', in the directed landscape. We show that disjoint optimizers always exist, and that their lengths can be used to construct an extended directed landscape. The extended directed landscape can be built from an independent collection of extended Airy sheets, which we define from the parabolic Airy line ensemble. We show that the extended directed landscape and disjoint optimizers are scaling limits of the corresponding objects in Brownian last passage percolation (LPP). As two consequences of this work, we show that one direction of the Robinson-Schensted-Knuth bijection passes to the KPZ limit, and we find a criterion for geodesic disjointness in the directed landscape that uses only a single parabolic Airy line ensemble. The proofs rely on a new notion of multi-point LPP across the parabolic Airy line ensemble, combinatorial properties of multi-point LPP, and probabilistic resampling ideas.
arXiv comment93 pages, 8 figures. To appear in Memoirs of the AMS
References in corpus (6)
- The KPZ fixed point
- The directed landscape
- Bulk properties of the Airy line ensemble
- Brownian absolute continuity of the KPZ fixed point with arbitrary initial condition
- Three-halves variation of geodesics in the directed landscape
- Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison