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Three-halves variation of geodesics in the directed landscape

Duncan Dauvergne, Sourav Sarkar, Bálint Virág

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.

Three-halves variation of geodesics in the directed landscape

Abstract

We show that geodesics in the directed landscape have -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- and that weight functions are not Hölder-, although these objects are known to be Hölder with all lower exponents.

Paper Structure

This paper contains 11 sections, 52 theorems, 219 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geodesic basics
  4. Brownian-Bessel decomposition
  5. Compressed exponential tails for displacement and length
  6. Environment around a geodesic
  7. Asymptotic independence of increments
  8. Uniform tail bounds for geodesic increments
  9. Variations
  10. Sharp upper bounds on Hölder exponents
  11. Open problems

Key Result

Theorem 1.1

Let $\pi$ be the unique geodesic from $(0,0) \to (0,1)$. As $\varepsilon\to 0$, let $t_\varepsilon\in (0,1)$ satisfy $\varepsilon^3/\min(t_\varepsilon, 1 - t_\varepsilon)\to 0.$ Then the environments around $(\pi,t_\varepsilon)$ at scale $\varepsilon$ satisfy as $\varepsilon \to 0$, where $B$ is a two-sided Brownian motion, $R$ is a two-sided Bessel-$3$ process, $\mathcal{L}$ is the part of a di

Figures (3)

  • Figure 1: Rescaled geodesics in the prelimiting Brownian model in Definition \ref{['d:DL']}. Each line is a Brownian motion with negative drift.
  • Figure 2: The main objects in Lemma \ref{['L:coal-2']}. For large $n$, with high probability all $\mathcal{L}^{I_n}$-geodesics to a point in $V_t$ will be close to the point $(\pi(s), s)$ at the time $s_n$ and hence contained in $U$.
  • Figure 3: An illustration of Proposition \ref{['p:coal-1']}. In the resampled landscapes $\mathcal{L}^{I_n}$, the geodesics to $(0, 1)$ or to points in $V$ will differ from the corresponding $\mathcal{L}$-geodesics, but only in a shrinking interval around $s$. The fact that the geodesics trees from $(0, 0)$ to $V \cup \{0, 1\}$ typically agree in $\mathcal{L}$ and $\mathcal{L}^{I_n}$ for large $n$ implies the local equality of the environments $Z_\varepsilon[\pi_n, t_\varepsilon]$ and $Z[\pi, t_\varepsilon]$ in Corollary \ref{['c:coal-2']}.

Theorems & Definitions (97)

  • Theorem 1.1: Brownian-Bessel decomposition
  • Theorem 1.2: Uniform tail bounds for geodesic increments
  • Theorem 1.3: Variations of geodesics and weights
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3: Lemma 10.2, DOV
  • Proposition 2.4: Proposition 10.5, DOV
  • Proposition 2.5: Corollary 10.7, DOV
  • Proposition 2.6: Proposition 12.3, DOV
  • Lemma 2.7
  • ...and 87 more