Let
$$
m=\max_x\mathbb E_xT_A
$$
and choose $x$ attaining the maximum. For every $t$, the at time $t$ gives
$$
m=\mathbb E_xT_A
\leq t+m\mathbb P_x(T_A>t).
$$
Thus
$$
\mathbb P_x(T_A\leq t)\leq\frac tm.
$$
If $t\geq t_{\mathrm{mix}}(1/4)$, then
$$
\mathbb P_x(X_t\in A)
\geq\pi(A)-\frac14\geq\frac14.
$$
Since $\{X_t\in A\}\subseteq\{T_A\leq t\}$, this is impossible when $t<m/4$. Allowing for integer times, one may take any smaller absolute constant, for example
$$
\boxed{t_{\mathrm{mix}}(1/4)\geq\frac18
\max_x\mathbb E_xT_A.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 215 / 4 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-215/4/b/solution