Markov chains of $Z$-oriented triangulations of surfaces

TL;DR

This paper studies Markov chains on $z$-oriented triangulations of surfaces, characterizing their ergodic behavior and linking it to Eulerian triangulation colorings.

Contribution

It introduces Markov chains for $z$-oriented triangulations and provides a characterization of their ergodicity, connecting to surface triangulation colorings.

Findings

Characterization of ergodicity of the Markov chains.

Connection between $z$-orientations and Eulerian triangulation colorings.

Insights into triangulation structures on surfaces.

Abstract

We consider triangulations of closed $2$-dimensional (not necessarily orientable) surfaces. Any minimal set of zigzags that double covers the set of edges provides a $z$-orientation of the triangulation. We introduce Markov chains of $z$-oriented triangulations. Our main result is a characterization of their ergodicity. This topic is closely connected to coloring of Eulerian triangulations.