Three-halves variation of geodesics in the directed landscape

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 has 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.

arXiv comment41 pages, 4 figures. Updated with the journal version of the paper

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Cited by in corpus (9)

• The stationary horizon and semi-infinite geodesics in the directed landscape
• Convergence of the Environment Seen from Geodesics in Exponential Last-Passage Percolation
• Disjoint optimizers and the directed landscape
• Local KPZ behavior under arbitrary scaling limits
• A Strong Duality Principle for Equivalence Couplings and Total Variation
• Temporal increments of the KPZ equation with general initial data
• Constructing Maximal Germ Couplings of Brownian Motions with Drift
• When the geodesic becomes rigid in the directed landscape
• Environment seen from infinite geodesics in Liouville Quantum Gravity