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2022 Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape
Erik Bates, Shirshendu Ganguly, Alan Hammond
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Electron. J. Probab. 27: 1-44 (2022). DOI: 10.1214/21-EJP706


Within the Kardar–Parisi–Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work [27] of Dauvergne, Ortmann, and Virág, this object was constructed and, upon a parabolic correction, shown to be the limit of one such model: Brownian last passage percolation. The limit object without parabolic correction, called the directed landscape, admits geodesic paths between any two space-time points (x,s) and (y,t) with s<t. In this article, we examine fractal properties of the set of these paths. Our main results concern exceptional endpoints admitting disjoint geodesics. First, we fix two distinct starting locations x1 and x2, and consider geodesics traveling (x1,0)(y,1) and (x2,0)(y,1). We prove that the set of yR for which these geodesics coalesce only at time 1 has Hausdorff dimension one-half. Second, we consider endpoints (x,0) and (y,1) between which there exist two geodesics intersecting only at times 0 and 1. We prove that the set of such (x,y)R2 also has Hausdorff dimension one-half. The proofs require several inputs of independent interest, including (i) connections to the so-called difference weight profile studied in [10]; and (ii) a tail estimate on the number of disjoint geodesics starting and ending in small intervals. The latter result extends the analogous estimate proved for the prelimiting model in [40].


Bálint Virág gave a seminar at the Rényi Institute in January 2019 after which he showed simulations of the measure μ from (1.13b) and indicated that the Hausdorff dimension of its support equals one-half. The third author attended this talk and acknowledges discussions with Bálint Virág in person and by email regarding the fractal geometry of various exceptional sets embedded in the directed landscape and relations between the measure μ and the Airy sheet. In these discussions, Bálint indicated an argument, due to him and Duncan Dauvergne, which may be used to prove that uniform convergence of geodesics in the directed landscape entails that all but finitely many intersect the limiting path. This fact is captured here by Theorem 1.18. The authors also thank Riddhipratim Basu, Timo Seppäläinen, Evan Sorensen, and Benedek Valkó for helpful discussions, and the referees for numerous useful suggestions.

E.B. was partially supported by NSF grant DMS-1902734. S.G. was partially supported by NSF grant DMS-1855688 and a Sloan Research Fellowship in Mathematics. A.H. was partially supported by NSF grant DMS-1855550 and a Miller Professorship at U.C. Berkeley.


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Erik Bates. Shirshendu Ganguly. Alan Hammond. "Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape." Electron. J. Probab. 27 1 - 44, 2022.


Received: 23 January 2021; Accepted: 15 September 2021; Published: 2022
First available in Project Euclid: 10 January 2022

MathSciNet: MR4361743
zbMATH: 1496.60115
Digital Object Identifier: 10.1214/21-EJP706

Primary: 60K35
Secondary: 28A80 , 60G15 , 60G57 , 60K37 , 82B44

Keywords: Airy sheet , Brownian last passage percolation , directed landscape , geodesics , Polymers

Vol.27 • 2022
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