Relaxation time ($t_{\mathrm{rel}}$)

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

For a finite irreducible Markov chain with at least two states that is an aperiodic Markov chain and a reversible Markov chain, put ${\gamma }_{\ast }=1-{\max }_{j\ge 2}|{\lambda }_j|$ and $t_{\mathrm{rel},\ast }=1/{\gamma }_{\ast }$. This is the reciprocal of the absolute L2 spectral gap of a reversible Markov chain. It agrees with the reciprocal of the ordinary spectral gap $1-{\lambda }_2$ for a lazy Markov chain, but need not agree when there are negative eigenvalues.

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 $r\le 1/2$,