The "largest" eigenvalue $1$ of a stochastic matrix is well-characterized by the classical [Perron-Frobenius theorem.][1] In particular, it gives sufficient conditions for the eigenvalue $1$ to be simple. 

Are there any sufficient conditions known when all (or at least all real) eigenvalues of such a matrix are simple? 

Thank you very much.


  [1]: https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem