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Geodesic networks and the disjointness gap in the directed landscape

Duncan Dauvergne, Oliver Scott Pankratz

Abstract

The directed landscape is a random directed metric on the plane that arises as the scaling limit of metric models in the KPZ universality class. For a pair of points p, q, the disjointness gap G(p; q) measures the shortfall when we optimize length over pairs of disjoint paths from p to q versus optimizing over all pairs of paths. Any spatial marginal of G is simply the gap between the top two lines in an Airy line ensemble. In this paper, we show that when the start and end time are fixed, the disjointness gap fully encodes the set of exceptional geodesic networks. The correspondence uses simple features of the disjointness gap, e.g. zeroes, local minima. We give a similar correspondence relating semi-infinite geodesic networks to a Busemann gap function. The proofs are deterministic given a list of soft properties related to the coalescent geometry of the directed landscape.

Geodesic networks and the disjointness gap in the directed landscape

Abstract

The directed landscape is a random directed metric on the plane that arises as the scaling limit of metric models in the KPZ universality class. For a pair of points p, q, the disjointness gap G(p; q) measures the shortfall when we optimize length over pairs of disjoint paths from p to q versus optimizing over all pairs of paths. Any spatial marginal of G is simply the gap between the top two lines in an Airy line ensemble. In this paper, we show that when the start and end time are fixed, the disjointness gap fully encodes the set of exceptional geodesic networks. The correspondence uses simple features of the disjointness gap, e.g. zeroes, local minima. We give a similar correspondence relating semi-infinite geodesic networks to a Busemann gap function. The proofs are deterministic given a list of soft properties related to the coalescent geometry of the directed landscape.
Paper Structure (20 sections, 53 theorems, 168 equations, 9 figures)

This paper contains 20 sections, 53 theorems, 168 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Applications
  3. The semi-infinite case
  4. Background
  5. The directed landscape
  6. The extended landscape
  7. Geodesics and optimizers
  8. Geodesic networks
  9. The almost sure set for Theorem \ref{['T: fin-time-thm']}
  10. Finite time networks I-III
  11. One-sided local minima
  12. Finite time networks IV-V
  13. Proofs of applications
  14. Preliminaries in the semi-infinite setting
  15. A correspondence for semi-infinite networks
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $Q_{x,y}^{-,+} = (-\infty, x) \times (y, \infty)$, $Q_{x,y}^{+,-} = (x, \infty) \times (-\infty, y)$, and let $Z:= G^{-1}(\{0\})$ be the zero set of the gap sheet. The following claims hold almost surely for all $x,y\in \mathbb{R}$.

Figures (9)

  • Figure 1: The 7 networks that can appear for a fixed time.
  • Figure 2: The $7$ possible semi-infinite network types. One gets the vertex 'at infinity' by connecting the paths at the top.
  • Figure 3: Possible fixed-time network to be ruled out.
  • Figure 4: Sketch of Lemma \ref{['L: seq-disj-geos']}.
  • Figure 5: An illustration of the creation of two bubbles. The left pictures is from the proof of Lemma \ref{['L: uses-l-or-r-bubble']}, and the right from the proof of Lemma \ref{['L: network-V-zero-restriction']}. The blue lines are the geodesic(s) for $(x-\epsilon_{n_k}, y+\delta_{n_k})$, and the black lines geodesics for $(x,0;y,1)$. The bubbles are indicated by the red dashed lines.
  • ...and 4 more figures

Theorems & Definitions (92)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Lemma 2.5
  • ...and 82 more