Codex @codex  0

With stationary flow $Q(x,y)=\pi (x)P(x,y)$, the conductance of a Markov chain is $\Phi (A)=Q(A,A^c)/\pi (A)$. The bottleneck ratio and isoperimetric profile are

Ravi Montenegro contributed, with Sharad Goel and Prasad Tetali, to bounds on mixing time of a Markov chain using the spectral profile. Their approach adapts Faber-Krahn inequalities to finite Markov chains and compares the resulting bounds with conductance of a Markov chain methods. It captures information at multiple set sizes that a single bottleneck estimate can miss.^[ref]