I know that many things are known when dealing with random walks on a finite (or even infinite) graph: mixing time, returns to origin, etc. All is based in the use of the Markovian property of such a random walk (I am assuming that on each vertex we can choose the next one among neighbours with uniform probability distribution).


My question is the following: when dealing with non-backtracking random walks (namely, we cannot go back through an edge we have just used, inducing on every edge a uniform distribution) we lose all markovian property, but this can be manage by taking orientation on edges. 


Q: is there some kind of 'universal' result for regular graphs concerning the existence of stationary distribution for such random walks? 


I have looked for bibliography on this topic (this should be like the first question on these models), but I have not been able to find any reference on this.



Non-backtracking random walk in regular (finite) graphs
Indeed, understanding non-backtracking walks is often the key to analyzing the simple random walk and random graphs. See e.g. [1], [2] and [3], [4]. 
Basic properties of the non-backtracking walk are collected in [5], Exercise 6.59.


[1] Lubetzky, Eyal, and Allan Sly. "Cutoff phenomena for random walks on random regular graphs." Duke Mathematical Journal 153, no. 3 (2010): 475-510.


[2] Lubetzky, Eyal, and Yuval Peres. "Cutoff on all Ramanujan graphs." Geometric and Functional Analysis 26, no. 4 (2016): 1190-1216.  

<https://www.math.nyu.edu/~eyal/papers/ramanujan.pdf>


[3] C. Bordenave. A new proof of Friedman’s second eigenvalue Theorem and its extension to random lifts. arXiv preprint arXiv:1502.04482, 2015.


[4] Bordenave, Charles, Marc Lelarge, and Laurent Massoulié. "Non-backtracking spectrum of random graphs: community detection and non-regular ramanujan graphs." In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pp. 1347-1357. IEEE, 2015.


[5] R. Lyons and Y. Peres. Probability on Trees and Networks. Cambridge University
Press. (2016). Available at <http://pages.iu.edu/~rdlyons/>.


