Markov chains of $Z$-oriented triangulations of surfaces

TL;DR

The paper studies Markov chains defined on $z$-oriented triangulations of closed surfaces, where a $z$-orientation is a minimal set of zigzags that double every edge. It characterizes ergodicity of the induced chain $oldsymbol{X}_{ au}$ in terms of edge/face types: the chain is not ergodic precisely when $(oldsymbol{ abla}, au)$ is $3$-colorable and all faces are of type II, with sphere and real projective plane admitting a simplified criterion. The authors show equivalences between aperiodicity, ergodicity, and the existence of a cycle whose length is not divisible by $3$, and they connect these properties to directed Eulerian triangulations arising from type II edges. They illustrate the theory with explicit examples (octahedral, toroidal grids, projective plane triangulations) and discuss how connected sums and a $T( abla)$ construction preserve or realize the different cases, providing a broad classification across surfaces. The work links zigzag geometry, Eulerian colorings, and stochastic dynamics on surface-embedded graphs, revealing a deep interplay between topological embedding, combinatorial coloring, and Markov-chain ergodicity.

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.

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Theorems & Definitions (21)

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• Theorem 1