Briefly: A hint (if this is easy), reference or derivation would be of great help.

## The question

Let $C_n$ be the directed cycle with loops in each of its $n$ vertices, and consider the random walk that at each step stays put with probability $\frac{1}{2}$ and moves clockwise with probability $\frac{1}{2}$.

Let $X_t$ be the state of the walk (i.e., the vertex visited) at time $t$, where $X_0=x$, for some vertex $x$ of $C_n$.

Given a parameter $\varepsilon$ and the vertex $x$, I am interested in bounding the value $t_0$ such that for all times $t$, with $t\geq t_0$ and all vertices $y$ of $C_n$

$$ \left|\Pr(X_t = y) - \frac{1}{n}\right| \leq \varepsilon. $$

## Some context

Let $C_n$ be an $n$-vertex undirected cycle. If the random walk on $C_n$ is ergodic (i.e., if $n$ is odd), then the stationary distribution is uniform (since $C_n$ is 2-regular). Furthermore, if $t=O(c n^2\log n)$, then $$|\Pr(X_t=y)-n^{-1}|<\exp(-c).$$

What i need to incorporate is the fact that every vertex has a loop (I do not care wether $n$ is even or odd), and that the walk is directed. Does this ruin the mixing time?