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
$$
\Phi_*=\inf_{0<\pi(A)\leq1/2}\Phi(A),
\qquad
\Phi_*(r)=\inf_{0<\pi(A)\leq r}\Phi(A),
\quad0<r\leq\frac12.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 215 / 2 / a / i / Solution

{title2=$\Phi(A)$}

For a <reversible Markov chain> with stationary flow $Q(x,y)=\pi(x)P(x,y)$, the conductance of a set $A$ is
$$
\Phi(A)=\frac{Q(A,A^c)}{\pi(A)}.
$$
Its <Cheeger constant> minimizes the boundary flow divided by the smaller stationary mass of the two sides of the cut.

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 Incoming links: Conductance of a Markov chain