The geometry of coalescing random walks, the Brownian web distance and KPZ universality
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.
Fields of Study & Subjects
Stochastic processes and statistical mechanics
Theoretical and Computational Physics
Complex Systems and Time Series Analysis
Publication details
- Journal / source
- The Annals of Probability
- Volume
- 54
- Issue
- 2
- Publication date
- 26 February 2026
- Language
- EN