Counter Examples to Invariant Circle Packing

TL;DR

This paper constructs a unimodular random planar triangulation without an invariant circle packing, disproving a previously posed problem, and also shows that a weaker form involving point-stationary circle packings is false through two different approaches.

Contribution

It provides the first counterexamples to the existence of invariant circle packings in unimodular random planar graphs and introduces methods using indistinguishability and finite approximations.

Findings

Counterexample to invariant circle packing existence

Disproof of the weaker point-stationary circle packing problem

Two distinct approaches for constructing counterexamples

Abstract

In this work, a unimodular random planar triangulation is constructed that has no invariant circle packing. This disputes a problem asked in [arXiv:1910.01614]. A natural weaker problem is the existence of point-stationary circle packings for a graph, which are circle packings that satisfy a certain mass transport principle. It is shown that the answer to this weaker problem is also false. Two examples are provided with two different approaches: Using indistinguishability and finite approximations.