The P\'olya Web

TL;DR

The paper introduces the Pólya Web, a novel coalescing random walk system based on Pólya urns, analyzing its geometric properties, convergence behavior, and extending results to a Yule Web variant.

Contribution

It presents the first analysis of a Pólya urn-based coalescing web, including negative association, convergence laws, and the extension to Yule processes.

Findings

Negative association of indicator variables proven using BKR inequality

Almost sure convergence of the number of components established

Extension of properties to the Yule Web through local scaling

Abstract

We introduce the P\'olya Web, a system of coalescing random walks based on the classic P\'olya urn model. This construction serves as an analogue to the web of coalescing random walks studied by T\'oth and Werner (1998), replacing simple symmetric random walks with P\'olya walks as primary constituents. First, we study the general web of up-right oriented coalescing random walks. We investigate its geometric properties and prove that certain indicator random variables satisfy negative association. Notably, the proof involves a non-trivial application of the van den Berg-Kesten-Reimer (BKR) inequality. Based on this property, we derive a strong law for the number of connected components generated by walks starting at the same time. Subsequently, we focus on the specific properties of the P\'olya Web. It is well-known that the normalized coordinates of a single P\'olya Walk converge almost surely to a beta-distributed random variable. We determine the joint distribution of these limiting variables in the coalescing framework. Using these joint densities, we provide exact calculations regarding the almost sure convergence of the number of components. Finally, by applying a local scaling to the P\'olya Web at the edges, we introduce the Yule Web, a web of coalescing Yule processes. We demonstrate that the fundamental properties and results derived for the P\'olya Web can be extended to this limiting case.