Zigzags in combinatorial tetrahedral chains and the associated Markov chain

TL;DR

This paper studies zigzags in random combinatorial tetrahedral chains, analyzing their distribution using Markov chains, and determines the limiting probabilities of chains having a specific number of zigzags as the chain length grows.

Contribution

It introduces a Markov chain model based on z-monodromies to analyze zigzags in tetrahedral chains and derives their asymptotic probabilities.

Findings

Maximum of 3 zigzags per chain up to reversal

Limiting probabilities for chains with 1, 2, or 3 zigzags

Use of Markov chain to model z-monodromies

Abstract

Zigzags in graphs embedded in surfaces are cyclic sequences of edges whose any two consecutive edges are different, have a common vertex and belong to the same face. We investigate zigzags in randomly constructed combinatorial tetrahedral chains. Every such chain contains at most $3$ zigzags up to reversing. The main result is the limit of the probability that a randomly constructed tetrahedral chain contains precisely $k\in\{1,2,3\}$ zigzags up to reversing as its length approaches infinity. Our key tool is the Markov chain whose states are types of $z$-monodromies.