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

The Dirichlet form of a Markov chain isWhen $P$ is reversible, this equals $⟨f,(I-P)g{⟩}_{\pi }$.

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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

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