Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 215 / 1

The laziness of $P=(Q+I)/2$ makes all its eigenvalues nonnegative. WriteThe relaxation time isIf $(f_i)$ is an orthonormal eigenbasis of $L^2(\pi )$ with $f_1=1$, the spectral decomposition is

On the diagonal,so every summand is nonnegative. Let $m=\lceil t_{\mathrm{rel}}\rceil$. For every $i\ge 2$,HenceMultiply by $\pi (x)f_i(x{)}^2$ and sum over $i\ge 2$ to obtain the required inequality.

The squared $L^2(\pi )$ distance has the diagonal identityThe diagonal excess is nonnegative and decreases with time. ThereforeUsing the identity supplied in the question givesAt $t=8{\mathbb{E}}_{\pi }{\tau }_x$ the right side is at most $1/16$, up to the immaterial integer rounding. Thus

Put $T=t_{\mathrm{mix}}^{(2)}(a,1/4)$ and $m=\lceil t_{\mathrm{rel}}\rceil$. The expected local time isPart c and the supplied return identity giveThe second term is bounded by the same infinite sum. Part b now yieldsThus the hinted universal constant $C=9e/(e-1)$ works.