I’m a noob in random walks theory on groups.
I don’t understand what does mixing time of a random walk is.
They say it is something about convergence to a steady distribution. Like all I understand so far about random walk is that if you are at a given position, there is a distribution on the set of generators that you will multiply next. So like that distribution is fixed.
So what do they mean by steady distribution?
Consider the random walk on the n-cycle, i.e., set $n \geq 2$ and $w_k = e^{i\frac{2k\pi}{n}}$ for all $k \in \{0, \dots, n\}$. Starting from $0$, you have $\frac{1}{2}$ chance to go in the direct sense, $\frac{1}{2}$ to go in the indirect sense.
It should be pretty clear that, after a very long time, you are in each position with probability $\frac{1}{n}$.
The mixing time is the time after which, starting from any distribution on the $n$-cycle, your distribution will be almost exactly uniform.
Interestingly there is often a cut-off phenomena, i.e. just before the mixing time your distribution is far from uniform, and right after this time your distribution is very close to uniform.