Recurrence of a Weighted Random Walk on a Circle Packing with Parabolic Carrier

TL;DR

This paper proves that a weighted random walk on a circle packing with a parabolic carrier is recurrent, extending the concept with a higher-dimensional analogue of Dubejko weights.

Contribution

It demonstrates recurrence of weighted random walks on circle packings with parabolic carriers and introduces a higher-dimensional analogue of Dubejko weights.

Findings

Weighted random walk is recurrent on parabolic circle packings.

Introduces a higher-dimensional analogue of Dubejko weights.

Extends recurrence results to new geometric settings.

Abstract

In this paper we show that given a circle packing of an infinite planar triangulation such that its carrier is parabolic, placing weights on the edges according to a certain natural way introduced by Dubejko, makes the random walk recurrent. We also propose a higher-dimensional analogue of the Dubejko weights.