Steady state distributions

Question

Suppose we have a discrete-state discrete-time Markov Chain with n states. We know that this Markov chain has a unique steady-state distribution. If you additionally know that the transition matrix $P \in \mathbb{R}^{n \times n}$ is doubly stochastic, i.e., both its rows and columns sum to 1, then what is that unique stead-state distribution?

Answer 1

If we have a doubly stochastic matrix, then we know that its steady state distribution will follow the uniform distribution. In other words, if we have $n$ states, then the steady state distribution is given by

$$\pi = \left(\underbrace{\frac{1}{n}, \frac{1}{n}, \ldots, \frac{1}{n}}_{n \text{ times}}\right). $$

A full proof is provided here: Help showing aMarkov chain with a doubly-stochastic matrix has uniform limiting distribution