Stationary-reset perturbation of a Markov chain

Let $P$ be a transition matrix with Stationary distribution $\pi$, and let $\Pi$ have every row equal to $\pi$. For $0\le a\le 1$, the stationary-reset perturbationfollows the original transition with Probability $1-a$ and independently resamples from stationarity with Probability $a$. Since $P\Pi =\Pi P=\Pi$ and ${\Pi }^2=\Pi$,The total variation identity is exact: subtracting stationarity multiplies every signed row difference by the same nonnegative factor. Once the first reset occurs the distribution remains stationary, so the factor also has a direct no-reset Probability interpretation.

This can destroy pre-cutoff. Take the dimension-$m$ Shift-refresh chain on a hypercube, whose distance is $d_m(t)=1-2^{t-m}$ for $t<m$ and zero for $t\ge m$. Use reset Probability $a_m=c/m$ for fixed $c>0$ and $m>c$. For $0<s<1$, ${\tilde{d}}_m(\lfloor sm\rfloor )\to e^{-cs}$, while at $s\ge 1$ the distance is zero. A positive fraction of the mixing scale therefore already causes a nonvanishing decrease from one. Any scale capable of making the later distance tend to zero must be at least order $m$, and its earlier fixed positive multiple cannot have distance tending to one. Thus this perturbed family lacks pre-cutoff even though the original has a sharp cutoff.