Three-halves variation of geodesics in the directed landscape

TL;DR

This paper investigates the fine-scale properties of geodesics in the directed landscape, revealing their 3/2-variation, cubic variation of weight functions, and asymptotic independence at small scales, with implications for their regularity.

Contribution

It introduces the 3/2-variation of geodesics and cubic variation of weight functions, along with small-scale limit descriptions involving Brownian-Bessel boundary conditions.

Findings

Geodesics have 3/2-variation.

Weight functions exhibit cubic variation.

Geodesic environments are asymptotically independent at small scales.

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 H\"older-$2/3$ and that weight functions are not H\"older-$1/3$, although these objects are known to be H\"older with all lower exponents.