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The geometry of coalescing random walks, the Brownian web distance and KPZ universality

Bálint Vető, Bálint Virág

TL;DR

The paper analyzes the geometry of coalescing random walks by introducing the discrete web distance $D^{\mathrm{RW}}$ and its scaling limit, the Brownian web distance $D^{\mathrm{Br}}$, defined via the Brownian web and its dual. It establishes convergence of $D^{\mathrm{RW}}$ to $D^{\mathrm{Br},\mathrm{LSC}}$ in the epigraph sense and reveals a rich structure of distance regions bounded by backward Brownian trajectories, with both continuous and discrete boundary descriptions. In a KPZ-type direction, the shear limit of the Brownian web distance connects to the directed landscape $\mathcal{L}$ and the Airy process, and the paper discusses extrapolations toward the full directed landscape and Airy sheet, as well as related objects like the Brownian castle. It also analyzes horizontal behavior, proving logarithmic scaling for horizontal displacements and detailing the continuity properties via Skorokhod reflection, thereby linking discrete models to universal KPZ-type objects. These results illuminate how scale-invariant, yet geometry-rich, distance metrics on random path ensembles exhibit both universal KPZ features and delicate boundary-driven structures.

Abstract

Coalescing simple random walks in the plane form an infinite tree. A natural directed distance on this tree is given by the number of jumps between branches when one is only allowed to move in one direction. The Brownian web distance is the scale-invariant limit of this directed metric. It is integer-valued and has scaling exponents 0:1:2 as compared to 1:2:3 in the KPZ world. However, we show that the shear limit of the Brownian web distance is still given by the Airy process. We conjecture that our limit theorem can be extended to the full directed landscape.

The geometry of coalescing random walks, the Brownian web distance and KPZ universality

TL;DR

The paper analyzes the geometry of coalescing random walks by introducing the discrete web distance and its scaling limit, the Brownian web distance , defined via the Brownian web and its dual. It establishes convergence of to in the epigraph sense and reveals a rich structure of distance regions bounded by backward Brownian trajectories, with both continuous and discrete boundary descriptions. In a KPZ-type direction, the shear limit of the Brownian web distance connects to the directed landscape and the Airy process, and the paper discusses extrapolations toward the full directed landscape and Airy sheet, as well as related objects like the Brownian castle. It also analyzes horizontal behavior, proving logarithmic scaling for horizontal displacements and detailing the continuity properties via Skorokhod reflection, thereby linking discrete models to universal KPZ-type objects. These results illuminate how scale-invariant, yet geometry-rich, distance metrics on random path ensembles exhibit both universal KPZ features and delicate boundary-driven structures.

Abstract

Coalescing simple random walks in the plane form an infinite tree. A natural directed distance on this tree is given by the number of jumps between branches when one is only allowed to move in one direction. The Brownian web distance is the scale-invariant limit of this directed metric. It is integer-valued and has scaling exponents 0:1:2 as compared to 1:2:3 in the KPZ world. However, we show that the shear limit of the Brownian web distance is still given by the Airy process. We conjecture that our limit theorem can be extended to the full directed landscape.
Paper Structure (17 sections, 25 theorems, 100 equations, 2 figures)

This paper contains 17 sections, 25 theorems, 100 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The discrete web distance
  3. The Brownian web distance
  4. KPZ universality
  5. The Brownian web and its dual
  6. Properties of the Brownian web distance
  7. Continuity properties of the Brownian web distance
  8. Left neighbourhoods of an interval
  9. Left neighbourhoods in the discrete web
  10. Convergence of discrete web distance to Brownian web distance
  11. The KPZ limit
  12. Discrete web distance in the horizontal direction
  13. Proofs of continuity properties
  14. Properties of continuous region boundaries
  15. Properties of discrete region boundaries
  16. ...and 2 more sections

Key Result

Proposition 1.3

For any $\alpha>0$, we have the equality in distribution

Figures (2)

  • Figure 1: Boundaries of left disks with radii $0$ (in blue), $1$ (in red) and $2$ (in green) for random walk and Brownian web distance. The centers of nontrivial left Brownian web disks are atypical points.
  • Figure 2: The asymptotic disk $t-\sqrt{t^2-x^2}\le 2$ for the shift term in Theorem \ref{['thm:DBetoTW']}

Theorems & Definitions (53)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Conjecture 1.9
  • Theorem 1.10
  • ...and 43 more