# Mixing time of random walks on graphs

Suppose that we start a lazy random walk on a connected graph. However, the starting node is picked from a distribution of $$\mu$$ and $$||\mu-\pi||_{TV}<1/8$$, where $$\pi$$ is the stationary distribution of the lazy random walk on the graph. Now I am interested to see if there is any result regarding the mixing time of this chain. For example, if we run the chain for $$t$$ steps rounds, what happens to the total variation distance of the chain from the stationary distribution.

For any vertex $$v$$, you can take $$\mu$$ to be $$7/8 \pi + 1/8 v$$, that is, the with probability $$1/8$$ pick $$v$$ and otherwise pick a random vertex according to $$\pi$$. Now the distance to stationarity at time $$t$$ will be $$1/8$$ of what it is when starting the chain at $$v$$. So it seems that there are really no new questions here to explore.