Spectral profile ( $Λ (r)$ )

The spectral profile minimizes a Dirichlet-form Rayleigh quotient over nonnegative functions supported on sets of stationary mass at most

r

.

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- Generalized Cheeger inequality Spectral profile

Generalized Cheeger inequality

For the isoperimetric profile

Φ_{*} (r)

of a reversible chain and

r \leq 1/2

,

\frac{Φ _{*} ( r ) ^{2}}{2} \leq Λ (r) \leq \frac{Φ _{*} ( r )}{1 - r} .

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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 215 / 3 / a / ii / Solution

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