Dirichlet form of a Markov chain ($\mathcal{E}(f,g)$)

For a reversible transition matrix $P$ with invariant distribution $\pi$,

For a transient weighted graph, $H_0$ is the completion of the finitely supported functions in the norm $\Vert f{\Vert }_{H_0}=\mathcal{E}(f,f{)}^{1/2}$. Its elements have finite Dirichlet energy and vanish at infinity in the energy sense.

The relaxation time of a finite reversible lazy Markov chain is the reciprocal of its spectral gap: $t_{\mathrm{rel}}=(1-{\lambda }_2{)}^{-1}$.

The spectral profile minimizes a Dirichlet-form Rayleigh quotient over nonnegative functions supported on sets of stationary mass at most $r$.

For the isoperimetric profile ${\Phi }_{\ast }(r)$ of a reversible chain and $r\le 1/2$,