The Pólya Web

Abstract: We introduce the Pólya Web, a system of coalescing random walks based on the classic Pólya urn model. This construction serves as an analogue to the web of coalescing random walks studied by Tóth and Werner (1998), replacing simple symmetric random walks with Pólya 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 Pólya Web. It is well-known that the normalized coordinates of a single Pólya 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 Pólya 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 Pólya Web can be extended to this limiting case.