I'm wanting to sample the discrete uniform distribution over $n = 10$ integers using MCMC. My question concerns the transition probability matrix, $P$. As I understand it, any symmetric, irreducible and aperiodic Markov Chain would converge to the uniform equilibrium distribution.
So I was wondering if there's any advantage in using some arbitrary symmetric matrix over the one that jumps to mind as the obvious choice which would be a $10\times10$ matrix with $1/10$ as every entry.
This seems a bit of an overkill, but if you do want to do it, you probably still want to minimize the autocorrelation, so you should have zeros on the diagonal and $1/9$ off the diagonal.