The two uniform infinite quadrangulations of the plane have the same law

TL;DR

This paper proves that two different constructions of uniform infinite random quadrangulations of the plane are statistically equivalent, establishing a key equivalence in their probabilistic laws.

Contribution

It demonstrates the equivalence in distribution of two distinct models of uniform infinite quadrangulations, unifying previous approaches.

Findings

Both models have the same probabilistic law.

The result bridges different constructions of infinite quadrangulations.

It simplifies the understanding of the structure of infinite quadrangulations.

Abstract

We prove that the uniform infinite random quadrangulations defined respectively by Chassaing-Durhuus and Krikun have the same distribution.