Spectral upper bound for total variation mixing
= Spectral upper bound for total variation mixing
For a finite transitive reversible chain with nontrivial eigenvalues $\lambda_i$,
$$
4\lVert P^t(x,\cdot)-\pi\rVert_{\mathrm{TV}}^2
\leq\sum_{i\geq2}\lambda_i^{2t}.
$$