 Spectral profile

The Spectral profile records the energy needed to concentrate a function on sets of a given stationary mass. For a finite irreducible Reversible Markov chain with Stationary distribution $\pi$, defineHere $\mathcal{E}$ is its Dirichlet form of a Markov chain, and $r\ge {\pi }_{\ast }:={\min }_x\pi (x)$ ensures that admissible nonempty sets exist. This is the variance-normalized convention.^[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 $a{c}^2/2$ 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 mass at most $r<1$, the Cauchy-Schwarz inequality gives ${\mathrm{Var}}_{\pi }(f)\ge (1-r)\Vert f{\Vert }_{L^2(\pi )}^2$, relating the two conventions within a factor $1/(1-r)$.