Zigzags in Combinatorial Tetrahedral Chains and the Associated Markov Chain

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 \(k\in \{1,2,3\}\) $k\in \{1,2,3\}$ zigzags up to reversing as its length approaches infinity. Our key tool is the Markov chain whose states are types of z-monodromies.