= Spectral profile
{title2=$\Lambda(r)$}
The <Spectral profile> records the energy needed to concentrate a <function> on sets of a given stationary mass. For a finite <irreducible Markov chain>[irreducible] <Reversible Markov chain> with <Stationary distribution> $\pi$, define
$$
\lambda(A)=\inf_{\substack{f\geq0,\ \operatorname{supp}f\subseteq A\\\operatorname{Var}_\pi(f)>0}}
\frac{\mathcal E(f,f)}{\operatorname{Var}_\pi(f)},\qquad
\Lambda(r)=\inf_{0<\pi(A)\leq r}\lambda(A).
$$
Here $\mathcal E$ is its <Dirichlet form of a Markov chain>, and $r\geq\pi_*:=\min_x\pi(x)$ ensures that admissible nonempty sets exist. This is the variance-normalized convention.<Mixing Time Bounds via the Spectral Profile by Sharad Goel and others>[Section 1.2, Definitions 1.4–1.5]{ref}
The profile is nonincreasing in $r$, because enlarging $r$ permits more supports. The ordinary <spectral gap> is a lower bound on every $\Lambda(r)$, but small supports can force a larger <Rayleigh quotient>. For the symmetric two-state chain with off-diagonal <Transition probability> $a>0$, a <function> supported on one state has energy $ac^2/2$ and <variance> $c^2/4$, so $\Lambda(1/2)=2a$.
The denominator matters. Replacing <variance> by $\sum_x\pi(x)f(x)^2$ gives a restricted Dirichlet eigenvalue instead. On a <support of a function>[support] of mass at most $r<1$, the <Cauchy-Schwarz inequality> gives $\operatorname{Var}_\pi(f)\geq(1-r)\lVert f\rVert_{L^2(\pi)}^2$, relating the two conventions within a factor $1/(1-r)$.
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