Solution
= Solution
The worst-case distance to stationarity is
$$
d(t)=\max_\sigma\|P^t(\sigma,\cdot)-\pi_\beta\|_{\rm TV}.
$$
The <Path coupling theorem> says that if a coupling contracts an integer-valued path metric by a factor $\alpha<1$ for every adjacent pair, then
$$
d(t)\leq\operatorname{diam}(\Omega)\alpha^t.
$$