Dynamics of Triangulations

TL;DR

This paper investigates Markov processes involving triangulation flips on a sphere, demonstrating their ergodicity and mixing properties, and exploring degree distributions that suggest a power law behavior.

Contribution

It introduces a natural example of triangulation flips that lacks detailed balance and analyzes the resulting degree distribution.

Findings

Processes are ergodic and mixing

An example without detailed balance is provided

Degree distribution appears to follow a power law d^{-4}

Abstract

We study a few problems related to Markov processes of flipping triangulations of the sphere. We show that these processes are ergodic and mixing, but find a natural example which does not satisfy detailed balance. In this example, the expected distribution of the degrees of the nodes seems to follow the power law $d^{-4}$.