Suppose that I have a random walk on the finite cyclic group of order $d > 2$, where the initial probability distribution $Q$ assigns the values $p, q, r$ to $-1, 0, 1$, respectively, where $p + q + r = 1$. I understand the behavior quite well when $p, q, r$ are taken to be all ${1\over3}$, but I don't understand it as well here.
For which values of $p, q, r$ does the corresponding random walk converge to the uniform distribution?
When we do have convergence, what is the rate of convergence?
For which values of $p, q, r$ is this rate maximized? Minimized?
This is just an answer to the first question on when the random walk converges to the uniform distribution.
See Theorem 2.1 and Proposition 2.3 in this article by Laurent Saloff-Coste:
Consider a left-invariant random walk on a finite group $G$, and let by $\Sigma \subset G$ the set of steps (right multiplications) which have non-zero probability.
(This is the situation in your case, and $\Sigma$ is the subset of $\{-1,0,1\} \subset \mathbb{Z}/n\mathbb{Z}$ corresponding to the elements for which the probability is non-zero.)
Then this random walk converges to the uniform distribution if:
1) $\Sigma$ generates $G$
and
2) $\Sigma$ is not contained in a coset of a proper normal subgroup of $G$
In the case of a finite cyclic group and $\Sigma \subset \{-1,0,1\}$, the conclusion would be that these hold as long as $\Sigma$ contains at least two elements unless $\Sigma = \{-1,1\}$ and the order of the group is even.