and seems not to be written anywhere for the case of finite graphs, although there is some overlap with the treatment of the special case in Example 2.3.1 of \[b\].SalCos:stflour/. For sharper bounds that depend on more information about the graph, see, e.g., Theorems 3.3.11 and 2.3.1 of \[b\].SalCos:stflour/ for the finite case and \[b\].BCG/ for the infinite case. See also \[b\].MP:evolve/.

l.rate Suppose that $Q$ is a transition matrix of a Markov chain that is reversible with respect to a positive measure $\pi$. If $\pi$ is finite, then we assume that $\pi$ is normalized to be a probability measure. Assume that $c := \inf \big\{ \pi(x) Q(x, y) \st x \ne y \hbox{ and }
Q(x, y) > 0 \big\} > 0$ and that $a := \inf_x Q(x, x) > 0$. For all states $x$ and all $k \ge 0$, we have $$\left|{Q^k(x, x) \over \pi(x)} - 1\right|
\le
\min \Big \{ {1 \over a c \sqrt{k+1}}\,,\  
{1 \over 2 a^2 c^2 (k+1)} \Big \}
\label e.finb$$ if $\pi$ is finite and $${Q^k(x, x) \over \pi(x)} \le {1 \over a c \sqrt{k+1}}
\label e.infinb$$ if $\pi$ is infinite.

Write $V$ for the state space and $E$ for the set 