Asymmetric Transition Probability Matrix with uniform stationary distribution

Question

I am solving a discrete Markov chain problem. For this I need a Markov chain whose stationary distribution is uniform(or near to uniform distribution) and transition probability matrix is asymmetric.

[ Markov chains like Metropolis hasting has uniform stationary distribution but transition probability matrix is symmetric ]

Answer 1

Take the matrix $A$ such that for every $i$, we have:

1. $A_{i,i} =0.5$
2. $A_{i,i+1}=0.5$ (for $i=n$, set $A_{n,0}=0.5$)

Then it is not symmetric, but has a uniform stationary distribution.