Properties of Random Directed Graphs with Fixed Out-Degree I am interested in properties of random directed graphs with fixed out-degree $d$.  I am imagining a random graph model where each vertex chooses d neighbors (say, with replacement) u.a.r.  


  Question: Is anything known about the stationary
  distribution and mixing times of random
  walks on these random graphs (for various
  values of $d$)?  


I am particularly interested in the case where $d = 2$, which corresponds to a model of random automata over a Boolean alphabet.  (Yes, I realize these graphs are often not connected, but what happens in a given component?)  I am happy with partial results and results about other properties of these graphs.

It seems most of the literature on random graphs focuses on the Erdős–Rényi model, which has very different properties than the model I am thinking about.
In the undirected case random $d$-regular graphs are expanders with high probability (not for $d=2$, but I think $d \ge 3$ suffices), which implies that the mixing time of random walks is $O(\log n)$.  I don't remember enough about these proofs to know whether everything goes through in the directed case (certainly some properties are different: the uniform distribution is no longer stationary), but it may be worth looking into.  Good references for expander graphs are Expander Graphs and their Applications by Hoory, Linial, and Wigderson and Pseudorandomness by Vadhan.