Given a finite, undirected, 2-connected graph $G = (V,E)$, a non-backtracking random walk on $G$ is a random walk on the vertices of $G$ that "cannot backtrack". That is, if the walk $P$ is of size $k$, then $P = (u_0, u_1, \ldots, u_{k-1})$ is such that $(u_i, u_{i+1}) \in E$ and $u_{i+2} \neq u_i$.
The question that I am interested in is the following: given some $S \subseteq V$, find $$ \lim_{k \rightarrow \infty} \Pr[u_k \in S] $$ That is, what is the probability that the random walk ends at a node from $S$?
In general, non-backtracking random walk is not a Markov chain, and I guess that there is no simple answer. I try to understand what happens in a specific case, but even that seems hard:
Suppose that $G$ is a cycle on $n$ vertices (so the vertex set is $[n]$ and there is an edge for all $(i, i+1)$ and $(0,n)$), and that we added an edge "in the middle", that is we add the edge $(0, \frac{n}{2})$. Suppose that $S = \{0, \frac{n}{2}\}$. Can we compute the probability that the non-backtracking random walk will be in $S$ after $k \rightarrow \infty$ steps?