Interchange process

Place one distinct label at each vertex of a finite connected weighted graph. At rate $c_{uv}=c_{vu}\ge 0$, interchange the labels at $u$ and $v$. This continuous-time Markov chain has a uniform distribution on a finite set as its stationary distribution. The position of one label is the random walk with the same rates.

On a finite connected weighted graph, the interchange process and the single-label random walk with the same symmetric rates have equal spectral gaps. This is a substantial theorem, not just the elementary inclusion of the single-label spectrum in the full spectrum. A proof is given in Proof of Aldous' spectral gap conjecture, Theorem 1.1.