Zigzags in combinatorial tetrahedral chains and the associated Markov chain

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.