Growing random planar network with oriented branching and fusion

TL;DR

This paper models a growing planar network with orthogonal branching and fusion, analyzing its long-term behavior and limiting distribution through a novel spatial branching process approach.

Contribution

It introduces a new model of a growing planar network with oriented branching and fusion, and provides explicit descriptions of its asymptotic distribution and convergence speed.

Findings

Long-term empirical measure converges after polynomial rescaling.

Explicit description of the limiting distribution and convergence speed.

The network's structure can be analyzed via a one-dimensional stick breaking model.

Abstract

We consider a growing planar network where a tip grows at constant speed, branches at constant rate and inactivates when it meets a branch already created. We only consider here orthogonal branching occurring always in the same direction. This yields a spatial branching property to the growing network. The connected components of the network then form a branching process of rectangles with double immigration. Using a spine approach for a typical rectangle and coupling arguments, the study is boiled down to a one dimensional stick breaking model with aging. We can then prove long time convergence of empirical measure of the family of rectangles after polynomial rescaling. The limiting distribution and speed of convergence can be explicitly described. The proofs also rely on the description of common ancestor of rectangles in the branching structure with double immigration.