Planar maps, random walks and circle packing

TL;DR

This paper discusses the interplay between planar maps, random walks, and circle packing, providing insights into their mathematical relationships and applications.

Contribution

It offers a comprehensive overview of the connections between planar maps, random walks, and circle packing, highlighting recent developments and open problems.

Findings

Circle packing provides a geometric representation of planar maps.

Random walks on planar maps exhibit specific recurrence properties.

The lecture notes synthesize key results and open questions in the field.

Abstract

These are lecture notes of the 48th Saint-Flour summer school, July 2018, on the topic of planar maps, random walks and the circle packing theorem.