= Peres product condition
{c}
{title2=$\gamma_n t_{\mathrm{mix}}^{(n)}\longrightarrow\infty$}
For finite <irreducible Markov chains> that are <aperiodic Markov chains> and <reversible Markov chains>, the <ratio definition of cutoff> and $T_n=t_{\mathrm{mix}}^{(n)}(1/4)\to\infty$ imply the Peres product condition, with $\gamma_n=1-\lambda_{2,n}$. If $\gamma_nT_n\leq B$ on a subsequence, then $\gamma_{*,n}\leq\gamma_n$ and the <eigenvalue lower bound for total variation mixing> give
$$
\frac{t_{\mathrm{mix}}^{(n)}(\varepsilon)}{T_n}\geq\left(\frac1B-\frac1{T_n}\right)\log\frac1{2\varepsilon}.
$$
For sufficiently small fixed $\varepsilon$, this contradicts the limit one forced by <cutoff for Markov chains>. The same argument proves the stronger conclusion $\gamma_{*,n}T_n\to\infty$.
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