The evolving-set identity $P^t(x,y)=\frac{\pi(y)}{\pi(x)}\mathbb P_{\{x\}}\{y\in S_t\}$

Let $P$ be a Markov chain on a finite state space $V$ with strictly positive stationary distribution $\pi$, and write $Q(S,y)=\sum_{x\in S}\pi(x)P(x,y)$ for the stationary flow from a set $S$ into a state $y$. The evolving-set process of Morris and Peres is the Markov chain on subsets of $V$ that, from the current set $S$, draws $u$ uniform on $(0,1]$ and passes to the superlevel set $\{y:Q(S,y)/\pi(y)\ge u\}$; its transition probability from $S$ to $T$ is the length of the interval of thresholds $u$ realizing $T$.

The theorem (Lemma 17.12 of Levin–Peres–Wilmer) asserts that the set process contains the original chain: for all states $x,y$ and every time $t$,

where the right-hand probability is over the evolving-set process started from the singleton $\{x\}$ — the sum of its $t$-step transition probabilities into the sets containing $y$.

Every question about $t$-step transition probabilities is thereby a question about how the random set grows and shrinks. Combined with the martingale property of $\pi(S_t)$ (the companion lemma), this identity is what converts martingale estimates on sets into the mixing and return-probability bounds of this mission.