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The stationary horizon and semi-infinite geodesics in the directed landscape

Ofer Busani, Timo Seppäläinen, Evan Sorensen

Abstract

The stationary horizon (SH) is a stochastic process of coupled Brownian motions indexed by their real-valued drifts. It was first introduced by the first author as the diffusive scaling limit of the Busemann process of exponential last-passage percolation. It was independently discovered as the Busemann process of Brownian last-passage percolation by the second and third authors. We show that SH is the unique invariant distribution and an attractor of the KPZ fixed point under conditions on the asymptotic spatial slopes. It follows that SH describes the Busemann process of the directed landscape. This gives control of semi-infinite geodesics simultaneously across all initial points and directions. The countable dense set $Ξ$ of directions of discontinuity of the Busemann process is the set of directions in which not all geodesics coalesce and in which there exist at least two distinct geodesics from each initial point. This creates two distinct families of coalescing geodesics in each $Ξ$ direction. In $Ξ$ directions, the Busemann difference profile is distributed like Brownian local time. We describe the point process of directions $ξ\inΞ$ and spatial locations where the $ξ\pm$ Busemann functions separate.

The stationary horizon and semi-infinite geodesics in the directed landscape

Abstract

The stationary horizon (SH) is a stochastic process of coupled Brownian motions indexed by their real-valued drifts. It was first introduced by the first author as the diffusive scaling limit of the Busemann process of exponential last-passage percolation. It was independently discovered as the Busemann process of Brownian last-passage percolation by the second and third authors. We show that SH is the unique invariant distribution and an attractor of the KPZ fixed point under conditions on the asymptotic spatial slopes. It follows that SH describes the Busemann process of the directed landscape. This gives control of semi-infinite geodesics simultaneously across all initial points and directions. The countable dense set of directions of discontinuity of the Busemann process is the set of directions in which not all geodesics coalesce and in which there exist at least two distinct geodesics from each initial point. This creates two distinct families of coalescing geodesics in each direction. In directions, the Busemann difference profile is distributed like Brownian local time. We describe the point process of directions and spatial locations where the Busemann functions separate.
Paper Structure (40 sections, 65 theorems, 302 equations, 9 figures)

This paper contains 40 sections, 65 theorems, 302 equations, 9 figures.

Table of Contents

  1. Introduction
  2. KPZ fixed point and directed landscape
  3. Semi-infinite geodesics and Busemann functions
  4. Stationary horizon as the Busemann process of the directed landscape
  5. Non-uniqueness of geodesics and random fractals
  6. Inputs
  7. Organization of the paper
  8. Acknowledgements
  9. Model and main theorems
  10. Notation
  11. Geodesics in the directed landscape
  12. KPZ fixed point
  13. Stationary horizon
  14. Semi-infinite geodesics
  15. Invariance and uniqueness of the stationary horizon under the KPZ fixed point
  16. ...and 25 more sections

Key Result

Theorem 2.1

Let $(\Omega,\mathcal{F},\mathbb P)$ be a probability space on which the stationary horizon $G=\{G_\xi\}_{\xi \in \mathbb{R}}$ and directed landscape $\mathcal{L}$ are defined, and such that the processes $\{\mathcal{L}(x,0;y,t):x,y \in \mathbb{R}, t > 0\}$ and $G$ are independent. For each $\xi \in (Invariance) For each $t > 0$, the equality in distribution $\{h_t({\space\raisebox{1.5pt}{\scaleob

Figures (9)

  • Figure 2.1: The stationary horizon. Each color represents a different parameter $\xi \in \{0,\pm 1,\pm 2,\pm 3,\pm 5,\pm 10\}$
  • Figure 2.2: On the left, a depiction of the non-uniqueness in Theorem \ref{['thm:DLSIG_main']}\ref{['itm:good_dir_coal']}: geodesics separate and coalesce back together, forming a bubble. After the first version of the present article was posted, Bhatia Bhatia-23 and Dauvergne Dauvergne-23 proved that this is the only possible configuration for this type of non-uniqueness--that is, geodesics which split and later coalesce can only split at the initial point. On the right, $\xi\in \Xi$. The blue/thin paths depict the $\xi -$ geodesics, while the red/thick paths depict the $\xi +$ geodesics. From each point, the $\xi -$ and $\xi +$ geodesics separate at points of $\mathfrak S$. The $\xi -$ and $\xi +$ families each have a coalescing structure.
  • Figure 5.1: Illustration of the proof of Lemma \ref{['lem:RM_geod_SIG']}. Here, the red/thick path denotes the path $\hat{\gamma}$ in the case $w_t < g(t)$, which is to the right of the rightmost geodesic between $(x,s)$ and $(g(u),u)$, which passes through $(w_t,t)$ by assumption. This gives the contradiction.
  • Figure 5.2: The blue/thin path represents $g_{(w,q_1)}^{\xi -,L}$ and the red/thick path represents $g$.
  • Figure 6.1: In this figure, $(x,s) \in \operatorname{NU}_0 \setminus \operatorname{NU}_1$ and $(y,t) \in \operatorname{NU}_1 \subseteq \operatorname{NU}_0$. It has since been shown by Bhatia Bhatia-23 and Dauvergne Dauvergne-23 that no such points $(x,s)$ exist.
  • ...and 4 more figures

Theorems & Definitions (149)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6: Busemann geodesics and general geodesics
  • Remark 2.7: Non-uniqueness of geodesics
  • Remark 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 139 more