Let $G(V, E)$ be a graph and
$\pi$ be a probability distribution on
V.
Denote by
$J$ the transition matrix of the simple random walk on
$G$.
Let $P$ be the transition matrix of the Metropolis chain with initial transition matrix $J$ and desired steady distribution $\pi$.
Given this information, is it always true that the metropolis chain will converge to $\pi$? I'm quite confused. Will a periodic graph (like the $4$-cycle) be a counterexample?