Lemmas 13.11 and 13.12 -- the variational characterization of the gap

Let $P$ be an irreducible Markov chain on a finite state space $V$ with at least two states, reversible with respect to its stationary distribution $\pi$ (detailed balance: $\pi(x)P(x,y)=\pi(y)P(y,x)$). Among the eigenvalues of $P$ — the real $\lambda$ admitting a nonzero $f$ with $Pf=\lambda f$ — let $\lambda_2$ be the largest eigenvalue different from $1$, and let $\gamma=1-\lambda_2$ be the spectral gap. For functions $f:V\to\mathbb R$ write $\mathbb E_\pi(f)=\sum_xf(x)\pi(x)$ for the mean, $\langle f,f\rangle_\pi=\sum_xf(x)^2\pi(x)$ for the squared $\ell^2(\pi)$-norm, and

for the Dirichlet form — the average squared local variation of $f$ along the chain's transitions.

The theorem (Lemmas 13.11 and 13.12 of Levin–Peres–Wilmer) asserts:

1. the spectral gap is the minimal Dirichlet energy over centred unit-norm functions: $\gamma=\inf\bigl\{\mathcal E(f)\;:\;\mathbb E_\pi(f)=0,\ \langle f,f\rangle_\pi=1\bigr\}$;
2. the infimum is attained by some such $f$ (an eigenfunction for $\lambda_2$).

This variational characterization lets one bound the gap from above by exhibiting any single test function, and from below by functional inequalities — the mechanism driving the Cheeger inequality and the comparison method of this mission.