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September 2022 Three-halves variation of geodesics in the directed landscape
Duncan Dauvergne, Sourav Sarkar, Bálint Virág
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Ann. Probab. 50(5): 1947-1985 (September 2022). DOI: 10.1214/22-AOP1574

Abstract

We show that geodesics in the directed landscape have 3/2-variation and that weight functions along the geodesics have cubic variation.

We show that the geodesic and its landscape environment around an interior point have a small-scale limit. This limit is given in terms of the directed landscape with Brownian–Bessel boundary conditions. The environments around different interior points are asymptotically independent.

We give tail bounds with optimal exponents for geodesic and weight function increments.

As an application of our results, we show that geodesics are not Hölder-2/3 and that weight functions are not Hölder-1/3, although these objects are known to be Hölder with all lower exponents.

Citation

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Duncan Dauvergne. Sourav Sarkar. Bálint Virág. "Three-halves variation of geodesics in the directed landscape." Ann. Probab. 50 (5) 1947 - 1985, September 2022. https://doi.org/10.1214/22-AOP1574

Information

Received: 1 November 2020; Revised: 1 December 2021; Published: September 2022
First available in Project Euclid: 24 August 2022

Digital Object Identifier: 10.1214/22-AOP1574

Subjects:
Primary: 60K35
Secondary: 82B23 , 82C22

Keywords: Airy sheet , Brownian–Bessel boundary , directed geodesic , directed landscape , Geodesic , KPZ , Last passage percolation , Scaling limit , variation

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 5 • September 2022
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