Scaling limit of the radial Poissonian web

TL;DR

This paper studies a radial Poissonian web model with independent jumps, demonstrating that its scaled structure converges to the Brownian Bridge Web, a collection of coalescing Brownian bridges, revealing its large-scale diffusive behavior.

Contribution

It introduces a new radial Poissonian web model with independent jumps and proves its convergence to the Brownian Bridge Web under diffusive scaling.

Findings

The scaled tree converges to the Brownian Bridge Web.

Paths in the model are independent jumps, differing from previous models.

The model exhibits diffusive behavior at large scales.

Abstract

We consider a variant of the radial spanning tree introduced by Baccelli and Bordenave. Like the original model, our model is a tree rooted at the origin, built on the realization of a planar Poisson point process. Unlike it, the paths of our model have independent jumps. We show that locally our diffusively rescaled tree, seen as the collection of the paths connecting its sites to the root, converges in distribution to the Brownian Bridge Web, which is roughly speaking a collection of coalescing Brownian bridges starting from all the points of a planar strip perpendicular to the time axis, and ending at the origin.