A view from infinity of the uniform infinite planar quadrangulation

TL;DR

This paper presents a simplified construction of the UIPQ using an extended tree mapping, confirming conjectures about its infinite geometry and analyzing geodesics.

Contribution

It introduces a new, simpler method for constructing the UIPQ by extending the Cori-Vauquelin-Schaeffer mapping without positivity constraints.

Findings

Proves Krikun's conjectures on the UIPQ's geometry at infinity

Provides a detailed analysis of infinite geodesics in the UIPQ

Simplifies the construction process of the UIPQ

Abstract

We introduce a new construction of the Uniform Infinite Planar Quadrangulation (UIPQ). Our approach is based on an extension of the Cori-Vauquelin-Schaeffer mapping in the context of infinite trees, in the spirit of previous work. However, we release the positivity constraint on the labels of trees which was imposed in these references, so that our construction is technically much simpler. This approach allows us to prove the conjectures of Krikun pertaining to the "geometry at infinity" of the UIPQ, and to derive new results about the UIPQ, among which a fine study of infinite geodesics.