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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-215/2/a/i/solution