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For a finite irreducible Reversible Markov chain with Stationary distribution $\pi$, let $Q(x,y)=\pi (x)P(x,y)$ and define its conductance profile byFor ${\pi }_{\ast }:={\min }_x\pi (x)\le r\le 1/2$ and the variance-normalized Spectral profile,The star here names the small-set profile on this restricted domain; some sources reserve a star for its extension truncated at $1/2$.^[ref]

For the upper bound, test the spectral-profile quotient with $f=1_A$. Its energy is $Q(A,A^c)$ and its variance is $\pi (A)(1-\pi (A))$. For the lower bound, take a nonnegative function supported on a set of mass at most $r$. Applying the level-set integral identity to $f^2$ bounds ${\Phi }_{\ast }(r)\Vert f{\Vert }_2^2$ by $\frac{1}{2}{\sum }_{x,y}Q(x,y)|f(x{)}^2-f(y{)}^2|$. Factoring the difference of squares and applying the Cauchy-Schwarz inequality bounds this last expression by $\sqrt{2\mathcal{E}(f,f)}\Vert f{\Vert }_2$. Squaring and using ${\mathrm{Var}}_{\pi }(f)\le \Vert f{\Vert }_2^2$ proves the lower bound. Thus a geometric obstruction to boundary flow produces an analytic obstruction to relaxation.

For a finite reversible lazy chain, the relaxation time is $t_{\mathrm{rel}}=1/\gamma$, where $\gamma =1-{\lambda }_2$ is the spectral gap. Its spectral profile isThe variational characterization of the spectral gap isEvery class $c_0^{+}(A)$ is contained in the class over which this last infimum is taken, so $\lambda (A)\ge \gamma$ and $\Lambda (r)\ge \gamma$. Therefore

Prasad Tetali coauthored the spectral profile method for bounding mixing time of a Markov chain with Sharad Goel and Ravi Montenegro. It refines a single spectral gap by considering functions supported on sets of different stationary masses. This makes the geometry of small sets useful for quantifying convergence to the stationary distribution.^[ref]

Ravi Montenegro contributed, with Sharad Goel and Prasad Tetali, to bounds on mixing time of a Markov chain using the spectral profile. Their approach adapts Faber-Krahn inequalities to finite Markov chains and compares the resulting bounds with conductance of a Markov chain methods. It captures information at multiple set sizes that a single bottleneck estimate can miss.^[ref]

Sharad Goel coauthored a method for estimating mixing time of a Markov chain through the spectral profile, with Ravi Montenegro and Prasad Tetali. Their analysis gives upper and lower convergence bounds and applies them to several families of graphs. The central mechanism is to track how relaxation changes with the stationary mass of a function's support.^[ref]

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 $1/(1-r)$.