For a reversible transition matrix $P$ on $L^2(\pi)$, the <spectral gap> is
$$
\gamma=1-\max\{\lambda:\lambda<1\text{ is an eigenvalue of }P\}.
$$
Equivalently for a lazy chain,
$$
\gamma=\inf_{\operatorname{Var}_\pi f>0}
\frac{\mathcal E_P(f,f)}{\operatorname{Var}_\pi f}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 215 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-215/3/a/solution